text:
\section{Introductory remarks}
\label{sec1}
\setcounter{equation}{0}

Explaining ferromagnetism from a simple model of itinerant electrons such as the standard Hubbard model
is a long-standing problem in the condensed matter theory.
Among many routes leading to ferromagnetism
the so-called flat-band ferromagnetism of Mielke and Tasaki\cite{mielke,tasaki,rev} is of special interest.
On the one hand,
many results for Mielke-Tasaki flat-band ferromagnetism have been obtained rigorously.
On the other hand,
this mechanism is important for material design,
since it opens interesting possibilities to obtain ferromagnetic materials in which magnetic atoms are completely missing.
In brief, the mechanism of this kind of ferromagnetism looks as follows.\cite{mielke,tasaki,rev}
Flat-band ground states
(i.e., the one-particle states from completely dispersionless band which is the lowest-energy one)
can be considered as one-particle states which are localized within small trapping cells on a lattice.\cite{lm,le,others_le}
Therefore,
exact many-electron ground states at low electron densities can be constructed simply by filling the traps.
Importantly,
in the case of connected (overlapping) traps,
electrons being in symmetric spin states avoid the on-site Hubbard
repulsion,
and, as a result, these states remain within the ground-state manifold for
$U>0$ with a $U$-independent energy.
Thus, the (degenerate) ground state consists of a set of ferromagnetic clusters.
If the electron density exceeds a threshold value,
a macroscopic wrapping ferromagnetic cluster appears
and ferromagnetism dominates the ground-state properties of thermodynamically large systems.\cite{mielke,tasaki,rev,le,myk1,myk2,ijmpb}
This ferromagnetism is robust against perturbation,
i.e., the ferromagnetic state remains stable for slightly perturbed models
which have a moderate change in the hopping integrals
leading to a slightly dispersive one-electron band.\cite{tasaki_jsp,kusakabe}

The above description of the emergence of ground-state ferromagnetism is based on the assumption,
that the trapping cells have common sites,
i.e., the so-called connectivity condition is satisfied for the localized one-electron states.
In other words,
the localized states overlap
and this was essential for the proofs in Refs.~\onlinecite{mielke,tasaki}.
On the other hand,
there are lattices which have lowest-energy flat bands but the traps do not have common sites
(nonoverlapping or isolated traps).
Those flat-band lattices
cannot support the above described mechanism for ferromagnetism,
since the trapped electrons cannot be in contact with each other, and, thus are unable to correlate.
Hence,  flat-band Hubbard models with isolated traps do not exhibit ferromagnetism at zero temperature,
rather  there is a macroscopically degenerate (i.e., the degeneracy
grows exponentially with the system size) ground-state manifold, where
paramagnetic states dominate.\cite{batista,prb2009,epjb2011,mielke2012}
However, the macroscopically degenerate ground-state
manifold is very sensitive to small perturbations which may lead  to subtle
effects of violations of the flat-band conditions.
This scenario has been investigated in Ref.~\onlinecite{oleg-johannes}
for the specific example of the frustrated diamond chain.
It was demonstrated
that the macroscopically degenerate ground-state manifold with all traps filled by electrons
results in a non-magnetic zero-temperature phase,\cite{oleg-johannes,prb2009}
but small deviations from the ideal flat-band geometry of hopping integrals
(which makes the flat band slightly dispersive)
lead to a fully polarized ferromagnetic many-electron ground state if $U>U_c$.
The value of $U_c$ depends on the strength of the deviation from the ideal geometry.
Note that another route to ground-state ferromagnetism without
connectivity condition in the flat band was discussed in
Ref.~\onlinecite{gkg}.

In the present paper we broaden and generalize our previous study
on the dispersion-driven ferromagnetism in flat-band Hubbard systems.\cite{oleg-johannes}
As already mentioned above, those studies referred to one particular lattice,
namely to an azurite-like\cite{azurite} diamond-Hubbard chain.
Moreover, analytical calculations presented in Ref.~\onlinecite{oleg-johannes}
were restricted to the fourth-order perturbation theory for a two-cell chain.
In the present study
we extend the analytical calculations to higher-orders perturbation theory
this way validating the previous results.
More importantly,
we consider other lattices with isolated trapping cells,
the one-dimensional ladder and the two-dimensional bilayer. These new
lattices have more degrees of freedom to constitute
 deviations from the ideal flat-band geometry.
Thus, we will demonstrate that the dispersion-driven ferromagnetism is a rather general mechanism
to establish ferromagnetic ground states in Hubbard models
having isolated trapping cells in the flat-band limit.
In addition to the analytical perturbation theory,
we also perform extensive exact-diagonalization studies.
Our analysis will, on the one hand, confirm the conclusions derived from the
study of the Hubbard diamond chain.\cite{oleg-johannes}
On the other hand, we will discuss further consequences
of deviations from the ideal flat-band geometry on ferromagnetism.
In particular,
we find that in some cases the required threshold on-site repulsion $U_c$ may be quite
small,
whereas in other cases ferromagnetic ground states do not appear at all.
There are also cases when ferromagnetic ground states appear only,
if the acquired bandwidth exceeds a threshold,
and then $U_c$ becomes a nonmonotonic function of the bandwidth.
Our findings are compactly collected in phase diagrams,
obtained both by analytical treatment and exact diagonalization,
which indicate the regions of dispersion-driven ground-state ferromagnetism.

The paper is organized as follows.
After a brief description of the models to be considered (Sec.~\ref{sec2}) and the methods to be used (Sec.~\ref{sec3})
we pass to a discussion of the obtained results for the diamond chain (Sec.~\ref{sec4}), the ladder (Sec.~\ref{sec5}),
and the bilayer (Sec.~\ref{sec6}).
We briefly summarize our results in Sec.~\ref{sec7}.
Several appendices present some lengthy formulas which are relevant for
the discussion in the main text of the paper.

\section{Models}
\label{sec2}
\setcounter{equation}{0}

We consider the standard repulsive one-orbital Hubbard model with the Hamiltonian
\begin{eqnarray}
\label{201}
H=\sum_{\sigma=\uparrow,\downarrow}H_{0,\sigma}+H_U,
\nonumber\\
H_{0,\sigma}=\sum_{(ij)}t_{ij}\left(c_{i,\sigma}^\dagger c_{j,\sigma}+c_{j,\sigma}^\dagger c_{i,\sigma}\right),
\;\;\;
t_{ij}>0,
\nonumber\\
H_U=U\sum_{i}n_{i,\uparrow}n_{i,\downarrow},
\;\;\;
U>0,
\end{eqnarray}
where generally accepted notations are used in Eq.~(\ref{201}).
We investigate the Hubbard model (\ref{201}) on two one-dimensional and one two-dimensional $N$-site lattices
which are shown in Fig.~\ref{f01},
namely the frustrated diamond chain, the frustrated two-leg ladder, and the frustrated bilayer.
In case of ideal flat-band geometry all hopping integrals $t_{ij}=t$ are
equal, except the hopping integral on the vertical bond $t_2$.
Then
one of the one-electron bands is strictly flat and it becomes the lowest one,
if $t_2$ is sufficiently large.
The localized-electron states are then located (trapped) on the vertical $t_2$-bonds.
Obviously, the trapping cells do not have common sites, the connectivity condition is violated,
and the zero-temperature state in the subspaces with $n\le{\cal{N}}$ electrons are nonmagnetic.
From Fig.~\ref{f01} it is obvious,
that the number of trapping cells ${\cal{N}}$ for the diamond chain and the ladder/bilayer
is ${\cal{N}}=N/3$ and ${\cal{N}}=N/2$, respectively.

\begin{figure
\begin{center}
\includegraphics[clip=on,width=72mm,angle=0]{figure01a.eps}\\
\vspace{5mm}
\includegraphics[clip=on,width=65mm,angle=0]{figure01b.eps}\\
\vspace{5mm}
\includegraphics[clip=on,width=75mm,angle=0]{figure01c.eps}\\
\vspace{5mm}
\caption{(Color online)
Lattices considered in the present paper:
The frustrated diamond chain, the frustrated two-leg ladder, and the frustrated bilayer
(from top to bottom).
The sites are enumerated by two indexes $m,i$:
The first one enumerates the cells,
$m=1,\ldots,{\cal{N}}$,
and
the second one enumerates the sites within a cell,
$i=1,2,3$ (diamond) and $i=1,2$ (ladder and bilayer).
The hopping integral for the vertical bond is $t_2$,
whereas the hopping integral along the bond connecting the sites $m,i$ and $m+1,j$ is denoted by $t_{ij}$,
see also the main text.
For ideal flat-band geometry $t_{ij}=t$ and $2t<t_2$ (diamond and ladder) or $4t<t_2$ (bilayer).}
\label{f01}
\end{center}
\end{figure}

We consider  deviations from the ideal flat-band geometry of the following
form: For the diamond chain, following Ref.~\onlinecite{oleg-johannes}, we
set $t_{13}=t_{32}=t_1\ne t_{23}=t_{31}=t_3$, $t_1+t_3=2t<t_2$ (azurite-like
geometry;\cite{azurite} for more general deformations see
Ref.~\onlinecite{fnt}).  It is convenient to parameterize the azurite-like
distortion as follows:
\begin{eqnarray}
\label{202}
t_1=t(1+\delta),
\;\;\;
t_3=t(1-\delta);
\nonumber\\
t=\frac{t_1+t_3}{2},
\;\;\;
\delta=\frac{t_1-t_3}{t_1+t_3}.
\end{eqnarray}
For the ladder/bilayer
$t_{11}$, $t_{12}$, $t_{21}$, and $t_{22}$ may be different, but we assume
$t_{11}+t_{12}+t_{21}+t_{22}=4t$ and $2t< t_2$ (ladder) or $4t< t_2$
(bilayer).  Again it is convenient to introduce the following
parameterization:
\begin{eqnarray}
\label{203}
t_{11}=t_l(1+\delta_l),
\;\;\;
t_{12}=t_f(1+\delta_f),
\nonumber\\
t_{21}=t_f(1-\delta_f),
\;\;\;
t_{22}=t_l(1-\delta_l);
\nonumber\\
t_l=\frac{t_{11}+t_{22}}{2},
\;\;\;
\delta_l=\frac{t_{11}-t_{22}}{t_{11}+t_{22}},
\nonumber\\
t_f=\frac{t_{12}+t_{21}}{2},
\;\;\;
\delta_f=\frac{t_{12}-t_{21}}{t_{12}+t_{21}}
\end{eqnarray}
with
$t_l+t_f=2t$.

In the distorted systems the lowest flat band with energy
$\varepsilon_1$ acquires a dispersion, i.e.,
$\varepsilon_1\to\varepsilon_1(\kappa)$,
resulting in a nonzero bandwidth $W_1>0$.
In Ref.~\onlinecite{oleg-johannes},
the acquired dispersion was characterized by a parameter $W_1/w_{2}$,
where $w_2$ denotes the bandwidth of the dispersive bands for the ideal flat-band geometry
(note that for the diamond chain there are two dispersive bands with identical bandwidth).
Furthermore,
for the diamond chain we have
$W_1\approx 2(t_3-t_1)^2/t_2$,
$w_{2}\approx 2(t_3+t_1)^2/t_2$
and therefore
$W_1/w_{2}\approx\Omega^2$,
where
$\Omega\equiv\left\vert(t_3-t_1)/(t_3+t_1)\right\vert$ used in Ref.~\onlinecite{oleg-johannes}
equals to $\vert\delta\vert$,
cf. Eq.~(\ref{202}).
However,
since for the Hubbard ladder/bilayer
the acquired bandwidth is not the only relevant parameter that controls the emergence of
ferromagnetism,
we prefer to use throughout this paper the above introduced parameters
$t$ and $\delta$ for the diamond chain
and
$t_l$, $t_f$, $\delta_l$, and $\delta_f$ for the ladder/bilayer.

\section{Methods}
\label{sec3}
\setcounter{equation}{0}

In our study we use an analytical perturbation-theory approach and numerical exact diagonalization.
Let us briefly explain these methods.
The starting point of the perturbation theory is
the splitting of the Hamiltonian $H$ of the problem at hand into the main part (unperturbed Hamiltonian) ${\sf{H}}_0$
and the perturbation ${\sf{V}}$,
i.e., $H={\sf{H}}_0+{\sf{V}}$.
Then we use the perturbation-theory formulas given in Ref.~\onlinecite{klein} (see also Appendix~A) to
determine the influence of the perturbation  ${\sf{V}}$ on the degenerate ground-state
manifold.
Since $t_2>0$ is the largest hopping integral and $U>0$,
the main part consists of the hopping terms on the vertical bonds and all on-site repulsion terms.
The perturbation consists of all other hopping terms.
Next we have to find all eigenstates and eigenvalues of the unperturbed Hamiltonian ${\sf{H}}_0$.
For $N$ sites and $n$ electrons there are altogether ${\cal{C}}_{2N}^n=(2N)!/[n!(2N-n)!]$
eigenstates.
For example, for $n={\cal{N}}=2,\,3,\,4,\,5$ ladder problems we have 28, 220, 1820, 15504 eigenstates, respectively.
In the considered regime, i.e., dominating positive $t_2$, $U>0$ is sufficiently large, and
$n={\cal{N}}$,
the ground state is $2^n$-fold degenerate,
i.e., 4-, 8-, 16-, 32-fold degenerate for $n={\cal{N}}=2,\,3,\,4,\,5$.
It has the form:
\begin{eqnarray}
\label{301}
\vert{\rm{GS}}\rangle=l^\dagger_{1,\sigma_1}\ldots l^\dagger_{n,\sigma_n}\vert{\rm{vac}}\rangle,
\nonumber\\
l^\dagger_{m,\sigma_m}
=\frac{1}{\sqrt{2}}\left(c^\dagger_{m,1,\sigma_m}-c^\dagger_{m,2,\sigma_m}\right).
\end{eqnarray}
The choice of the concrete linear combinations of states (\ref{301})
used as a starting point of perturbation theory
is related to the model with
perturbation.
Supposing an effective magnetic Heisenberg model for the low-energy degrees
of freedom,\cite{oleg-johannes} the choice
of ground states of the unperturbed Hamiltonian ${\sf{H}}_0$ which account
the SU(2) symmetry
of the Hubbard Hamiltonian is straightforward, for more details see
Appendix~B.
The resulting perturbation-theory formulas up to the sixth order are collected in Appendix~A
(see also Appendices~C, D, and E).
It is in order to mention here,
that in the small-$U$ limit, in addition to the states (\ref{301}), also states with two electrons in one cell,
become relevant.
As a result, the perturbation theory starting from the set of states (\ref{301}) may fail for $U \to 0$, see below.

To perform the fourth and sixth order perturbation theory
we use the symbolic computation software {\it Mathematica}.
To implement the symbolic calculation we used the SNEG package,
see Ref.~\onlinecite{sneg},
for Mathematica.
The package handles the non-commutative multiplication of,
e.g.,
fermionic creation and annihilation operators.
This is required to perform the perturbation theory in higher order for larger Hubbard
clusters.
For a compact sketch of the procedure see Appendix~F.

For the numerical exact diagonalization we use  J.~Schulenburg's {\it spinpack}.\cite{spinpack1,spinpack2}
This code allows the calculation of the ground state for the Hubbard model with a half-filled lowest band
up to $N=20$ sites.
Thus, by considering various system sizes the finite-effects can be estimated.
The comparison of the results obtained by  two different approaches
finally allows to get a consistent description of the ground-state phases
of the considered Hubbard systems.

\section{Diamond chain}
\label{sec4}
\setcounter{equation}{0}

The Hubbard model Hamiltonian on the diamond chain is given in Eq.~(\ref{201})
with the following explicit form for $H_{0,\sigma}$:
\begin{eqnarray}
\label{401}
H_{0,\sigma}=\sum_{m}\left[
t_{2}c_{m,1,\sigma}^\dagger c_{m,2,\sigma}
\right.
\nonumber\\
\left.
+t_{1}\left(c_{m,1,\sigma}^\dagger c_{m,3,\sigma}+c_{m,3,\sigma}^\dagger c_{m+1,2,\sigma}\right)
\right.
\nonumber\\
\left.
+t_{3}\left(c_{m,2,\sigma}^\dagger c_{m,3,\sigma}+c_{m,3,\sigma}^\dagger c_{m+1,1,\sigma}\right)
+{\rm{H.c.}}\right],
\end{eqnarray}
see Fig.~\ref{f01}.
Eq.~(\ref{401}) corresponds to an azurite-like deformation.\cite{azurite}
Furthermore, we assume half filling of the lowest nearly flat one-electron band,
i.e., the number of electrons equals the number of cells $n={\cal{N}}$.

Extensive exact-diagonalization calculations for this model were reported in
Ref.~\onlinecite{oleg-johannes}.
However,
the analytical treatment by perturbation theory was restricted to fourth-order
calculations
for the two-cell diamond chain with open boundary conditions consisting of
$N=5$ sites. (Note, that for the special diamond-chain geometry
the second-order perturbation theory is not sufficient to describe
ground-state ferromagnetism.\cite{oleg-johannes})
 In this paper we present  the sixth-order perturbation theory  and consider also a
larger cluster  consisting of three cells in fourth-order perturbation theory.
That allows to validate the previous lower-order approach and promises a
better agreement with  exact diagonalization for larger deviations from the
ideal flat-band geometry.

The results
for the triplet and singlet energies calculated for the cluster of $N=5$ sites with $n=2$ electrons up to the sixth order,
\begin{eqnarray}
\label{402}
E_t=-2t_2+E^{(2)}+E_t^{(4)}+E_t^{(6)}+\ldots,
\nonumber\\
E_s(U)=-2t_2+E^{(2)}+E_s^{(4)}(U)+E_s^{(6)}(U)+\ldots,
\end{eqnarray}
are given in Appendix~C.
From the obtained data one can see
that with increasing of the order of perturbation-theory calculations
the analytical results for the triplet and singlet energies monotonically approach the exact-diagonalization data from above.
The critical on-site repulsion
 $U_c$ is determined from the equation $E_t=E_s(U_c)$.
In fourth-order perturbation-theory we get a compact formula\cite{oleg-johannes}
\begin{eqnarray}
\label{403}
\frac{U^{(4)}_c}{t_2}=\frac{\sqrt{16+65\delta^{2}}+9\vert\delta\vert}{1-\delta^{2}}\vert\delta\vert.
\end{eqnarray}
Eq.~(\ref{403}) implies that in fourth order $U_c/t_2$ depends only on the deviation from
the ideal flat-band geometry controlled by $\delta$,
but not on $t$ or $t_2$.
Unfortunately,
in sixth order $U^{(6)}_c$ obtained as a solution of the equation
$E_t^{(4)}+E_t^{(6)}=E_s^{(4)}(U^{(6)}_c)+E_s^{(6)}(U^{(6)}_c)$
has to be calculated numerically, and cannot be presented in a compact
analytical form.
By contrast to  $U^{(4)}_c$, the sixth-order result  $U^{(6)}_c/t_2$ weakly
depends on $t_2$, which was also found in our exact-diagonalization results.
The corresponding results for $U^{(4)}_c$ and $U^{(6)}_c$ are shown in
Fig.~\ref{f02}.
It is evident, that the difference between the values of $U_c^{(4)}$ and $U_c^{(6)}$ at least for small $\delta$, where the perturbation theory is valid, is small
(the difference in Fig.~\ref{f02} becomes only visible if $\delta$ exceeds 0.4).
Thus, we confirm that the simple equation (\ref{403}) describes the phase
boundary surprisingly well.

Another way to extend the previous perturbation-theory calculations of
Ref.~\onlinecite{oleg-johannes} is to enlarge the cluster sizes used for the
perturbation theory. For that we consider $n=3$ electrons on the three-cell diamond chain with open boundary conditions which has $N=8$ sites.
Already in fourth order the perturbation theory becomes more ambitious,
since we have to take into account much more states, see Appendix~C.
Remarkably, for the larger cluster we get the same value of $U^{(4)}_c$ as given in Eq.~(\ref{403}).

\begin{figure
\begin{center}
\includegraphics[clip=on,width=62.5mm,angle=0]{figure02.eps}
\caption{(Color online)
Phase diagram for the Hubbard diamond chain.
Ferromagnetic ground states appear for $U>U_c$.
$U_c$ is shown as a function of $\delta$, $t=1$, see Eq.~(\ref{202}).
The various critical lines $U_c(\delta)$ are obtained by perturbation theory and exact diagonalization.}
\label{f02}
\end{center}
\end{figure}

Our results  are summarized in Fig.~\ref{f02},
where we also show some exact-diagonalization results obtained earlier.\cite{oleg-johannes}
This figure provides evidence, that
the sixth-order perturbation-theory calculations ($N=5$) almost do not change the predictions for $U_c(\delta)$ according to Eq.~(\ref{403}),
although there is a weak dependence of $U_c/t_2$ on $t_2$ in agreement with exact-diagonalization data
(compare the curves PT6 for $t_2=3$ and $t_2=6$ in Fig.~\ref{f02}).
The fact that Eq.~(\ref{403}) has been obtained now from calculations for both two-cell and three-cell diamond chains
(i.e., for $N=5$ and $N=8$), also
explains the  good agreement of Eq.~(\ref{403}) with exact-diagonalization results for longer chains
(e.g., for ${\cal{N}}=6$ cells, see Fig.~\ref{f02}).
Finally,
we emphasize again that our new results demonstrate that the formula for
$U_c$ given in Eq.~(\ref{403}) provides a simple and sufficiently precise criteria
for emergence of ground-state ferromagnetism in the Hubbard diamond chain.

\section{Ladder}
\label{sec5}
\setcounter{equation}{0}

Next we consider as a new example for a flat-band model with isolated
trapping cells the Hubbard model on a frustrated ladder, see
Fig.~\ref{f01}.
We point out at the beginning that, by contrast to the diamond chain, there is
no intermediate site between two trapping cells.
The explicit form for $H_{0,\sigma}$ in Eq.~(\ref{201}) is
\begin{eqnarray}
\label{501}
H_{0,\sigma}=\sum_{m}\left(
t_{2}c_{m,1,\sigma}^\dagger c_{m,2,\sigma} \qquad \qquad
\right.
\nonumber\\
\left.
+t_{11}c_{m,1,\sigma}^\dagger c_{m+1,1,\sigma}
+t_{12}c_{m,1,\sigma}^\dagger c_{m+1,2,\sigma}
\right.
\nonumber\\
\left.
+t_{21}c_{m,2,\sigma}^\dagger c_{m+1,1,\sigma}
+t_{22}c_{m,2,\sigma}^\dagger c_{m+1,2,\sigma}
+{\rm{H.c.}}\right),
\end{eqnarray}
see Fig.~\ref{f01}.

Using the notations of Eq.~(\ref{203}),
the one-electron dispersion relations for this model can be written in a compact manner as follows:
\begin{eqnarray}
\label{502}
\varepsilon_{1,2}(\kappa)
=2t_l\cos\kappa \qquad \qquad \qquad \qquad \qquad \qquad
\nonumber\\
\mp\sqrt{\left(t_2+2t_f\cos\kappa\right)^2+ 4t_l^2\delta_l^2\cos^2\kappa+4t_f^2\delta_f^2\sin^2\kappa}.
\end{eqnarray}
Flat-band geometry occurs
when $t_{11}=t_{12}=t_{21}=t_{22}=t$ or $t_l=t_f=t$, $\delta_l=\delta_f=0$ and $2t<t_2$.
Then
$\varepsilon_{1}(\kappa)=\varepsilon_{1}=-t_2$
and
$\varepsilon_{2}(\kappa)=t_2+4t\cos\kappa > \varepsilon_{1}$.

We consider a quite general deviation from the ideal flat-band geometry,
and assume only that $t_{11}+t_{12}+t_{21}+t_{22}=4t$ or $t_l+t_f=2t$ and $2t<t_2$.
Thus after fixing $t_l$ and $t_f$ with the restriction $t_l+t_f=2t<t_2$
we are left with two free parameters, $\delta_l$ and $\delta_f$ [see
Eq.~(\ref{203})], constituting a two-dimensional  parameter region.
Except the general case of deformations, we will also
consider two special deformations,
(i) a symmetric deformation
with $t_{11}=t_{22}$, $t_{12}=t_{21}$ and $t_{11} \ne t_{12}$ ($t_l\ne t_f$, $\delta_l=\delta_f=0$)
and
(ii)
a semi-symmetric deformation
with $t_{11}=t_{12}$, $t_{21}=t_{22}$ and $t_{11} \ne t_{21}$ ($t_l=t_f=t$, $\delta_l=\delta_f=\delta\ne
0$) which is identical to
$t_{11}=t_{21}$, $t_{12}=t_{22}$ and $t_{11} \ne t_{12}$ ($t_l=t_f=t$, $\delta_l=-\delta_f=\delta\ne 0$),
since all results depend only on $\delta_{l}^2$ and $\delta_{f}^2$,
see, e.g., Eq.~(\ref{502}).
For case (i) the dispersion relation  Eq.~(\ref{502}) becomes
\begin{eqnarray}
\label{503}
\varepsilon_{1,2}(\kappa)=\mp t_2 + 2\left(t_{l} \mp t_{f}\right)\cos\kappa,
\end{eqnarray}
whereas for case (ii) translates  into
\begin{eqnarray}
\label{504}
\varepsilon_{1,2}(\kappa) = 2t\cos\kappa\mp\sqrt{\left(t_2+2t\cos\kappa\right)^2+ 4t^2\delta^2}.
\end{eqnarray}
It is worth noting
that the acquired bandwidth of the former flat band due to the symmetric deformation may be larger than due to the semi-symmetric one.
On the other hand,
while the symmetric deformation does not lead to ferromagnetic ground states
at all,
see below,
the semi-symmetric one produces ferromagnetic ground states for very small $U>U_c$,
see below.
Obviously, the acquired bandwidth as the only relevant
parameter is insufficient to characterize the capability to obtain ground-state ferromagnetism.

In what follows
we first discuss perturbation-theory results in comparison with exact-diagonalization data for ladders up to ${\cal{N}}=4$ cells ($N=8$ sites)
and then present all analytical findings along with exact diagonalization for $N=12,\,16,\,20$ (${\cal{N}}=6,\,8,\,10$) in phase diagrams.

\subsection{Two electrons and two cells}
\label{ladder_two_cells}

We begin with the case of $n=2$ electrons on the ladder of ${\cal{N}}=2$ cells with open boundary conditions imposed.
Perturbation-theory calculations for the energies of the triplet state and the singlet state can be easily
obtained by symbolic computation up to the sixth order:
\begin{eqnarray}
\label{505}
E_t=-2t_2    +E_t^{(2)}+E_t^{(4)}+E_t^{(6)}+\ldots,
\nonumber\\
E_s(U)=-2t_2 +E_s^{(2)}(U)+E_s^{(4)}(U)+E_s^{(6)}(U)+\ldots .
\end{eqnarray}
Here the second-order corrections are as follows:
\begin{eqnarray}
\label{506}
E_t^{(2)}=-\frac{t_l^2\delta_l^2+t_f^2\delta_f^2}{t_2},
\nonumber\\
E_s^{(2)}(U)
=
-\frac{(t_l-t_f)^2}{t_2}
-2\frac{t_l^2\delta_l^2+t_f^2\delta_f^2}{2t_2+U}
-\frac{8(t_l-t_f)^2}{U}.
\end{eqnarray}
The explicit lengthy expressions for the higher-order corrections are given in Appendix~D.
Typical dependences of low-lying energies on $U$ are shown in
Figs.~\ref{f03}(a), \ref{f03}(b), and \ref{f03}(c)
for a particular general deformation, a symmetric deformation, and a semi-symmetric deformation, respectively.

\begin{figure
\begin{center}
\includegraphics[clip=on,width=78mm,angle=0]{figure03a.eps}\\
\includegraphics[clip=on,width=78mm,angle=0]{figure03b.eps}\\
\includegraphics[clip=on,width=78mm,angle=0]{figure03c.eps}
\caption{(Color online)
Energies of low-lying states (triplet -- red, singlet -- blue) as a function of on-site repulsion $U$
(perturbation theory up to sixth order and exact-diagonalization data)
for $n=2$ electrons on the ladder of ${\cal{N}}=2$ cells (open boundary conditions).
(a)
$t_2=3$, $t_{11}=0.85$, $t_{12}=0.95$, $t_{21}=1$, $t_{22}=1.2$ (general deformation).
(b)
$t_2=3$, $t_{11}=t_{22}=1.1$, $t_{12}=t_{21}=0.9$ (symmetric deformation).
(c)
$t_2=3$, $t_{11}=t_{21}=1.1$, $t_{12}=t_{22}=0.9$ (semi-symmetric deformation);
exact diagonalization yields $U_c \approx  0.015$,
whereas the perturbation-theory prediction is $U_c^{(6)}=0$.}
\label{f03}
\end{center}
\end{figure}

The conclusions obtained from the formulas and plots (Fig.~\ref{f03}) of the singlet and
triplet energies are as follows:
In the small-$U$ limit the perturbation theory may fail, cf. Figs.~\ref{f03}(a) and \ref{f03}(b).
The reason for this has been mentioned above already:
In the small-$U$ limit some relevant excited states approach the ground-state manifold.
The deviation from the ideal flat-band geometry
leads to more drastic effects and
also to a larger diversity in the energy dependence on $U$ than for the diamond chain considered in the previous section.
The behavior of $E_t$ and $E_s(U)$ shown in Fig.~\ref{f03}(a) for the general
case qualitatively
resembles that for the diamond chain (cf. Fig.~\ref{f08} in Appendix~C).
On the other hand,
the symmetric and semi-symmetric cases are totally unlike.
Namely,
as long as the perturbation theory converges, for the symmetric deformation, case (i), the singlet energy
(circles and blue curves)
is always lower than the triplet energy
(triangles and red curves), $E_s<E_t=-2t_2$, see Fig.~\ref{f03}(b).
Note that all exact-diagonalization data also yield
$E_s<E_t$ for the symmetric case.
For the semi-symmetric case the triplet energy becomes the lowest one,
$E_t<E_s(U)$,
if $U$ exceeds a very small critical value $U_c$,
see Fig.~\ref{f03}(c).
[For the case shown in Fig.~\ref{f03}(c)
exact diagonalization gives $U_c\approx 0.015$
and
the perturbation-theory result is $U_c^{(6)}=0$.]
That means,
ferromagnetism does not appear at all  for the symmetric deformation,
whereas for the semi-symmetric case only a very small $U$ is required to promote its appearance.
Next important difference in comparison to the diamond-chain case is related to the energy scale
(compare Figs.~\ref{f03} and \ref{f08}):
The splitting of triplet and singlet for the ladder occurs already in the second order
(and only in the fourth order for the diamond chain).
This can be traced back to the difference in lattice geometries.
Thus, for the ladder  the second-order  perturbation theory already provides useful results.

The above described features of the energy dependences on $U$
can be understood by a more detailed analysis of the perturbation-theory
treatment, see Appendices~A and B.
For that we consider the action of the perturbation ${\sf V}$ on the triplet and singlet
states, i.e.,
${\sf V}\vert t,\pm 1\rangle$, ${\sf V}\vert t,0\rangle$, and ${\sf V}\vert s\rangle$.
The results depend on the symmetry of the imposed deformation.
Thus,
for the symmetric case
${\sf V}\vert t\rangle=0$,
but
${\sf V}\vert s\rangle\propto (l^\dagger_{a,\uparrow}l^\dagger_{a,\downarrow}+l^\dagger_{b,\uparrow}l^\dagger_{b,\downarrow})\vert {\rm{vac}}\rangle$.
As a consequence,
the unperturbed triplet energy $-2t_2$ remains unchanged after switching on ${\sf
V}$,
whereas the unperturbed singlet energy $-2t_2$ decreases after switching on ${\sf V}$
and ferromagnetism cannot arise.
Moreover,
the state ${\sf V}\vert s\rangle$ overlaps with ``dangerous'' excited states of ${\sf{H}}_0$
(which contain $l^\dagger_{a,\uparrow}l^\dagger_{a,\downarrow}$, $l^\dagger_{b,\uparrow}l^\dagger_{b,\downarrow}$
and have the energy $-2t_2+U$ for $U\to 0$)
leading to the failure of the perturbation theory in the small-$U$ limit.
On the other hand, for the semi-symmetric case
${\sf{V}}\vert t\rangle$
contains
$c_{m,1,\sigma}^\dagger c_{m,2,\sigma}^\dagger\vert{\rm{vac}}\rangle$
or
$(c_{m,1,\uparrow}^\dagger c_{m,2,\downarrow}^\dagger + c_{m,1,\downarrow}^\dagger
c_{m,2,\uparrow}^\dagger)\vert{\rm{vac}}\rangle$,
whereas
${\sf{V}}\vert s\rangle\propto (c_{m,1,\uparrow}^\dagger c_{m,1,\downarrow}^\dagger - c_{m,2,\uparrow}^\dagger c_{m,2,\downarrow}^\dagger)\vert{\rm{vac}}\rangle$.
Since the state
$(c_{m,1,\uparrow}^\dagger c_{m,1,\downarrow}^\dagger - c_{m,2,\uparrow}^\dagger c_{m,2,\downarrow}^\dagger)\vert{\rm{vac}}\rangle$
is orthogonal to the dangerous excited states of ${\sf{H}}_0$,
the perturbation theory does not fail in the small-$U$ limit.
Moreover,
the states ${\sf{V}}\vert t\rangle$ and ${\sf{V}}\vert s\rangle$ have the same overlap integral with the excited states of ${\sf{H}}_0$ with the energies 0 and $U$, respectively.
Therefore, the decrease of the triplet energy exceeds the decrease of the singlet energy
instantaneously as $U>0$,
i.e., ferromagnetism appears for infinitesimally small positive $U$.

In second order the perturbation theory yields a compact formula
for the critical value of on-site repulsion $U_c$.
Using Eq.~(\ref{506}) we get
\begin{eqnarray}
\label{507}
\frac{U_c^{(2)}}{t_2}
=
\frac{5\vert t_l-t_f\vert +\sqrt{9(t_l-t_f)^2+16(t_l^2\delta_l^2+t_f^2\delta_f^2)}}{-(t_l-t_f)^2+t_l^2\delta_l^2+t_f^2\delta_f^2}
\nonumber\\
\times\vert t_l-t_f\vert. \qquad
\end{eqnarray}
Obviously,
for symmetric deformations,
when $t_l\ne t_f$ and $\delta_l=\delta_f=0$,
Eq.~(\ref{507}) gives for $U_c^{(2)}=-8t_2 <0$,
that is consistent with  the absence of  ferromagnetism in this case.
It is also obvious, that formula (\ref{507})
yields $U^{(2)}_c=0$ for $t_l=t_f$,
i.e., for $t_{11}-t_{12}-t_{21}+t_{22}=0$.
That criterion,  $t_l=t_f$,  holds
for semi-symmetric deformations, where in addition also $\delta_l=\delta_f$
is valid.
However, in higher-order perturbation theory as well as in exact diagonalization
we find that the constraint  $t_l=t_f$ does not imply $U_c=0$, rather $U_c$
may become large for the general case $\delta_l\ne \delta_f$, if
$\delta_l$ or $\delta_f$ become of the order of unity, see Fig.~\ref{f06}.

\begin{figure
\begin{center}
\includegraphics[clip=on,width=72mm,angle=0]{figure04.eps}
\caption{(Color online)
There is no ferromagnetic ground states for the Hubbard ladder
in the region around the origin of the plane $\delta_l$--$\delta_f$
[$t_{11}-t_{12}-t_{21}+t_{22}=2(t_l-t_f)\ne 0$].
Analytical predictions based on the second-order perturbation-theory calculations (\ref{507})
(lines)
are compared with exact-diagonalization data for $N=16$, $t_2=3$
(symbols)
for several values of $t_l$ and $t_f$, $t_l+t_f=2$.}
\label{f04}
\end{center}
\end{figure}

Supposing that the energies behave smoothly as changing deformations,
we can expect that there is a finite parameter region in the vicinity of
the symmetric case without ground-state ferromagnetism.
Indeed, for $t_l\ne t_f$  the second-order formula (\ref{507}) leads to an elliptic shape
in the  $\delta_l$--$\delta_f$ plane given by
\begin{eqnarray}
\label{508}
\left(\frac{t_l}{t_l-t_f}\delta_l\right)^2+\left(\frac{t_f}{t_l-t_f}\delta_f\right)^2=1.
\end{eqnarray}
We illustrate this behavior in Fig.~\ref{f04}, where we also show a few
points obtained by exact diagonalization which are in qualitative agreement
with the predictions from Eq.~(\ref{508}).
It is worthwhile to remark
that Eq.~(\ref{508}) remains unaltered if interchanging
$t_l \leftrightarrow t_f$ and $\delta_l \leftrightarrow \delta_f$
(this symmetry is also evident in Fig.~\ref{f04}).
However,
exact-diagonalization data shown by symbols in Fig.~\ref{f04} do not show this symmetry present in the second-order results,
i.e., it is not generally  present in the model,
cf., e.g., Eq.~(\ref{502}).

\subsection{Three (four) electrons and three (four) cells}

Let us discuss briefly the perturbation theory for larger clusters.
In the case of three electrons on the ladder of three cells we face a $2^3$-fold degenerate ground
state,
which consists of the quadruplet $\vert q\rangle$ (total spin is 3/2) and two doublets $\vert d1\rangle$ and $\vert d2\rangle$ (total spin is 1/2).
We are interested in the energies $E_q$, $E_{d1}$, and $E_{d2}$.
In Appendix~D,
we provide explicit expressions for these energies
\begin{eqnarray}
\label{509}
E_q=-3t_2+E_q^{(2)}+E_q^{(4)}+\ldots,
\nonumber\\
E_{di}(U)=-3t_2+E_{di}^{(2)}(U)+E_{di}^{(4)}(U)+\ldots,
\nonumber\\
i=1,2.
\end{eqnarray}
In the case of four electrons on the ladder of four cells we face a $2^4$-fold degenerate ground state,
which consists of the quintuplet $\vert Q\rangle$ (total spin is 2),
three triplets $\vert t1\rangle$, $\vert t2\rangle$, $\vert t3\rangle$ (total spin is 1),
and two singlets $\vert s1\rangle$, $\vert s2\rangle$ (total spin is 0).
In Appendix~D,
we provide explicit expressions for their energies
\begin{eqnarray}
\label{510}
E_Q=-4t_2+E_Q^{(2)}+E_Q^{(4)}+\ldots,
\nonumber\\
E_{ti}(U)=-4t_2+E_{ti}^{(2)}(U) +E_{ti}^{(4)}(U)+\ldots,
\nonumber\\
i=1,2,3,
\nonumber\\
E_{sj}(U)=-4t_2+E_{sj}^{(2)}(U) +E_{sj}^{(4)}(U)+\ldots,
\nonumber\\
j=1,2.
\end{eqnarray}
We report corresponding results for the energies up to the fourth order along with exact-diagonalization data
for the general, symmetric, and semi-symmetric deformations
for $n={\cal{N}}=3$ and $n={\cal{N}}=4$ in Appendix~D.
The main features of these results resemble strongly the ones discussed in the previous subsection for $n={\cal{N}}=2$.
Therefore, the main conclusions obtained from those data for the energies of
larger cells are
consistent with those discussed in Sec.~\ref{ladder_two_cells} for two
cells.
Most remarkably,
within the second-order perturbation theory,
the critical value $U_c^{(2)}$ for the three-cell and four-cell clusters
coincide with $U_c^{(2)}$ for the two-cell cluster,
i.e., it is given by Eq.~(\ref{507}).

Let us finally mention that within the perturbation theory for ${\cal{N}}=4$ cells
the fully polarized ferromagnetic state (it is a quintuplet for ${\cal{N}}=4$)
is in competition with triplet and singlet states.
We find,
cf. Fig.~\ref{f11},
that either a singlet or the ferromagnetic quintuplet is the ground state.
This finding,
that the fully polarized ferromagnetic state competes with a nonmagnetic singlet state
(but not with partially polarized states)
is supported by exact-diagonalization data obtained for systems with an even number of cells ${\cal{N}}>4$.

\subsection{Phase diagram}

In this subsection we collect analytical and numerical findings
to construct the ground-state phase diagrams of the Hubbard ladder.
According to Eq.~(\ref{203}),
there are three parameters which characterize the ladder,
i.e., $t_l$ and $t_f$ with $t_l+t_f=2t<t_2$, $\delta_l$, and $\delta_f$.
We set $t_2=3$, $t_l+t_f=2$.
After fixing  $t_l$ and $t_f$ we are left with two free parameters $\delta_l$ and $\delta_f$.
We consider the first quadrant of positive $\delta_l$ and $\delta_f$ in the $\delta_l$--$\delta_f$ plane.
We move through the quadrant by straight lines
in the horizontal direction ($\delta_f$ is fixed, $\delta_l$ varies),
in the vertical direction ($\delta_l$ is fixed, $\delta_f$ varies),
as well as along the diagonal $\delta_l=\delta_f=\delta$.
Certainly perturbation-theory results are reasonable only for small deviations from the ideal flat-band geometry.
However, there are no such restrictions for exact-diagonalization data.

We begin with a quite general case assuming $t_l=1.025$, $t_f=0.975$ and $\delta_l=0$.
The dependence of $U_c$ on $\delta_f$ is reported in Fig.~\ref{f05}.
The ground state is ferromagnetic above the curves $U_c(\delta_f)$;
this region is denoted as FM.
In this case,
the dependence of $U_c$ on the acquired bandwidth is a nonmonotonic function:
For small $\delta_f$ ferromagnetism does not appear at all
[in agreement with Eq.~(\ref{508})];
increasing  $\delta_f$ beyond a threshold value $\delta_{f1}$
ferromagnetism sets in and $U_c$ decreases with growing $\delta_f$.
Second-order perturbation theory, Eq.~(\ref{508}), predicts
$\delta_{f1}\approx 0.051$,
exact diagonalization for $N=16$ yields
$\delta_{f1}\approx 0.053$.
Beyond $\delta_f\approx 0.4$ the critical repulsion  $U_c$ starts to increase with increasing of $\delta_f$.
This behavior is obtained from both the fourth-order perturbation theory and exact diagonalization
for different system sizes with open and/or periodic boundary conditions imposed.
The second-order perturbation theory gives qualitatively correct results only for small $\delta_f<0.4$.
From exact-diagonalization data for $N=16$ it is obvious
that there is  again a threshold value $\delta_{f2}$
(for  $N=16$ we found $\delta_{f2} \approx 3.25$)
above which no ferromagnetism appears.
Fig.~\ref{f05} illustrates a quite subtle interplay of the hopping-integral geometry and the on-site Hubbard repulsion
required for establishing of ground-state ferromagnetism.

\begin{figure
\begin{center}
\includegraphics[clip=on,width=62.5mm,angle=0]{figure05.eps}
\caption{(Color online)
Phase diagram in the quarter plane $\delta_f$ -- $U/t_2$
for the ladder with $t_2=3$, $t_l=1.025$, $t_f=0.975$, and $\delta_l=0$
obtained by perturbation-theory calculations and by exact diagonalization
for $N=6,\;8,\;12,\;16$ with open and/or periodic boundary conditions.}
\label{f05}
\end{center}
\end{figure}

Next we pass to the case $t_l=t_f=1$.
The dependences of $U_c$ on $\delta_f$, on $\delta_l=\delta_f=\delta$, and on $\delta_l$
are reported in panels (a), (b), and (c) in Fig.~\ref{f06}, respectively.
The ground state is ferromagnetic above the curves $U_c(\delta)$;
this region is denoted as FM.
We recall that in the case $t_l=t_f$ from Eq.~(\ref{507}) we get $U^{(2)}_c=0$;
nonzero values of $U_c$ come only from higher-order (in fact, fourth-order) calculations.
Furthermore,
for the semi-symmetric deformation,
i.e., $\delta_l=\delta_f=\delta$,
the perturbation theory yields $U_c^{(4)}=0$.
Obviously, higher-order processes should lead to finite values for $U_c$,
as it is indicated by the exact-diagonalization data shown Fig.~\ref{f06}(b).

As can be seen in Figs.~\ref{f06}(a)  and \ref{f06}(c),
analytical results which refer to the case of ${\cal{N}}=3,\,4$ cells with open boundary conditions
and
exact-diagonalization data which refer to the case of ${\cal{N}}=6,\,8,\,10$ cells
are in a reasonable agreement.
By contrast to the parameter situation shown in Fig.~\ref{f05},
in all cases presented in Fig.~\ref{f06}
ground-state ferromagnetism can be obtained also for small deviations from the flat-band geometry
(controlled by $\delta_f$ and/or $\delta_l$).
Comparing the exact-diagonalization data for different system sizes $N$
we observe that the finite-size effects remain small,
thus  the discussed phenomenon should be present for thermodynamically large systems, too.

It is in order to mention a special finite-size effect
that may appear  for large values of $\delta_l$ and/or  $\delta_f$.
In this limit, the dominating hopping parameters may correspond to geometries
which do not fit to the initial ladder structure.
Thus, for $t_{11}=1+\delta_l$, $t_{22}=1-\delta_l$ and small $\delta_f$,
in the limit of $\delta_l\to\infty$ the legs of the ladder form two almost decoupled chains.
Such a finite simple Hubbard chain  at quarter filling with an odd number of electrons
(i.e., a chain of 6 or 10 sites with 3 or 5 electrons)
has a ferromagnetic ground state.
Therefore, the limit of large deviations, shown for completeness  in our
figures, goes beyond the primary focus of discussing the dispersion-driven
ferromagnetism in systems with ladder geometry.

\begin{figure
\begin{center}
\includegraphics[clip=on,width=62.5mm,angle=0]{figure06a.eps}\\
\includegraphics[clip=on,width=62.5mm,angle=0]{figure06b.eps}\\
\includegraphics[clip=on,width=62.5mm,angle=0]{figure06c.eps}
\caption{(Color online)
Phase diagram in the quarter plane $\delta$ -- $U/t_2$
for the ladder with $t_2=3$, $t_l=t_f=1$
obtained by fourth-order perturbation theory
and by exact diagonalization
(note that second-order perturbation theory yields $U^{(2)}_c=0$).
(a)
$\delta_l=0,\;0.05$.
(b)
$\delta_l=\delta_f=\delta$;
fourth-order perturbation-theory calculations yield zero value for $U_c$.
(c)
$\delta_f=0,\;0.05$.}
\label{f06}
\end{center}
\end{figure}

\section{Bilayer}
\label{sec6}
\setcounter{equation}{0}

As mentioned already, the mechanism leading  to the emergence of ferromagnetism
driven by kinetic energy is not restricted to dimension $D=1$.
To illustrate this, we consider the two-dimensional counterpart of the
Hubbard ladder, namely the frustrated bilayer,
see Fig.~\ref{f01}.
From the technical point of view, the two-dimensional model is more
challenging, since the smallest cluster appropriate for perturbation theory  and imaging the basic geometry of the
bilayer is built
by five cells (a central cell with four neighboring cells).
Furthermore, in contrast to the ladder for the exact diagonalization we do not have a
sequence of finite lattices of $N=12,16,20$ sites in $D=2$. The smallest finite
bilayer lattice
with periodic boundary conditions has $N=16$ sites.
Hence, we cannot provide a detailed discussion of the bilayer model, rather
we will demonstrate for a particular parameter set that the mechanism of kinetic-energy-driven
ferromagnetism also holds in $D=2$.

In analogy to the ladder,
for the bilayer one of the two one-electron bands is flat if
$t_{11}=t_{12}=t_{21}=t_{22}=t$ or $t_l=t_f=t$, $\delta_l=\delta_f=0$
and it becomes the lowest one if
$4t<t_2$.
Within fourth-order perturbation theory we are able to calculate
the energies of the fully polarized sextuplet (total spin 5/2)
 and of the quadruplets (total spin 3/2),
\begin{eqnarray}
E_S=-5t_2+E_S^{(2)}+E_S^{(4)}+\ldots,
\nonumber\\
E_{qi}(U)=-5t_2+E_{qi}^{(2)}(U)+E_{qi}^{(4)}(U)+\ldots,
\nonumber\\
i=1,2,3,4,
\end{eqnarray}
see Appendix~E.
Hence, our perturbation-theory treatment remains incomplete,
since we cannot compare with the energies of the five doublets with total spin 1/2.
On the other hand,
the comparison with the exact-diagonalization data for the five-cell cluster,
where the doublet states are taken into account,
yields an excellent agreement between both approaches.
That is because for this cluster the level crossing between the sextuplet and the lowest quadruplet
takes place at the same $U$ as for the crossing of  sextuplet and the lowest doublet.

As a first (remarkable) outcome we find,
that the second-order result $U_c^{(2)}$ again is given by Eq.~(\ref{507}).
We show numerical data for the critical repulsion $U_c$  for the set of parameters
$t_2=5$, $t_l=1.025$, $t_f=0.975$, and $\delta_l=0$
in the ground-state phase diagram
presented  in Fig.~\ref{f07} (cf. the corresponding phase diagram for the ladder shown
Fig.~\ref{f05}).

Basically  the same features as for the corresponding ladder are also found
for the phase diagram of the bilayer.
However, it is obvious that $U_c$ for the finite lattice of $N=16$ sites with periodic boundary conditions
is noticeably above perturbation-theory results and the exact-diagonalization results for $N=10$ sites.
We argue that the finite system of $N=10$ sites with open boundary conditions
is only a very rough model of thermodynamically large bilayer,
since only one (among five) vertical bond has the same environment as in infinite lattice.
The finite system of $N=16$ sites with periodic boundary conditions is free of this shortcoming.

\begin{figure
\begin{center}
\includegraphics[clip=on,width=62.5mm,angle=0]{figure07.eps}
\caption{(Color online)
Phase diagram in the quarter plane $\delta_f$ -- $U/t_2$
for the bilayer with $t_2=5$, $t_l=1.025$, $t_f=0.975$, and $\delta_l=0$
obtained by perturbation-theory calculations and by exact diagonalization
for $N=10,\,16$ with open and periodic boundary conditions.}
\label{f07}
\end{center}
\end{figure}

\section{Conclusions}
\label{sec7}
\setcounter{equation}{0}

We have used perturbation theory as well as exact diagonalization of
finite systems
to examine the kinetic-energy-driven emergence of ferromagnetic ground states
in Hubbard models with a half-filled lowest-energy flat band for lattices
which do not obey the connectivity condition (isolated trapping cells).
Generally speaking,
if
(i) the flat band acquires a small dispersion this way allowing to the previously localized electrons to correlate
and
(ii) the on-site Hubbard repulsion $U$ is sufficiently strong
the ground state becomes ferromagnetic.
However,
the relation between the required $U_c$ and the acquired bandwidth might be quite
intricate.
Thus,
for some deformation geometries ferromagnetism does not appear at all,
for others it appears already for small $U$;
in some cases $U_c$ is an increasing function of the deformation strength,
whereas in others it becomes nonmonotonic.
The mechanism leading to kinetic-energy-driven emergence of ferromagnetism
is studied in detail for one-dimensional systems with isolated trapping cells.
However, as it is demonstrated for a specific two-dimensional system this mechanism
works in higher dimensions as well.
Although our analysis refers to finite systems,
the observed finite-size behavior indicates convincingly that such a scenario should survive in the thermodynamic
limit, too.
Thus our main conclusion is
that the described phenomenon is a quite general way of establishing ground-state ferromagnetism
in the repulsive Hubbard model at low electron densities
around the flat-band limit.

Furthermore,
for special examples,
the diamond chain, the ladder as well as the bilayer,
we have obtained simple analytical formulas, cf. Eq.~(\ref{403}) and Eq.~(\ref{507}),
which amazingly well estimate the region of ground-state ferromagnetism.
From the technical point of view,
we have elaborated computer-adapted scheme for analytical perturbation-theory calculations up to the sixth order.

Finally,
it is in order to notice
that experimental searches for Mielke-Tasaki flat-band ferromagnetism remain an ambitious goal of numerous experimental studies,
see, e.g., Refs.~\onlinecite{ijmpb,experiments}.
Our findings offer new perspectives for investigating solid-state realization of flat-band
ferromagnetism, since the emergence of ferromagnetism in systems with isolated
trapping cells does not require  fine tuning of
parameters, rather it can be found in a quite wide parameter region.

\section*{Acknowledgments}

The present study was supported by the DFG (project RI615/21-1).
O.~D. acknowledges the kind hospitality of the University of Magdeburg in April-May and October-December of 2015.
O.~D. would like to thank the Abdus Salam International Centre for Theoretical Physics (Trieste, Italy)
for partial support of this study through the Senior Associate award.

redpajama_set_name: RedPajamaArXiv


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Company spokesperson Della DiPietro advised the move is effective March 28.
"The construction industry has taken a big hit and we had a significant loss of revenue last year, and we, as many in the market, expect that the severely depressed construction industry will continue to suffer in the future," DiPietro told USGNN.com™ this afternoon.
The company's automotive business will not be affected by the closing of the commercial line, DiPietro said.
"That plant has fabrication operations for automotive glass and those will continue," she said. "The plant's not shutting down."
One industry source, who wished to remain anonymous, commented on the announcement, "One of Zeledyne's challenges was that glass manufacturers are moving to more energy-efficient offerings including solar, and that requires a huge investment that wasn't feasible in this economy."
The anonymous source added, "There is some speculation that this positions Zeledyne for a possible sale of its automotive glass holdings. Zeledyne has one ace in its hand with its Ford contracts, which would make a sale attractive. The other thing this does is it reduces capacity in some areas, which in the long run might make for a healthier industry."
Zeledyne announced its purchase of Ford Motor Company Automotive Components Holdings (ACH) glass business in April 2008. CLICK HERE for related story.
Stay tuned to USGNN.com™ for updates as they are available.

redpajama_set_name: RedPajamaCommonCrawl


FIBER! What an exciting topic right? I know you are all chopping at the bit just waiting for me to get into the exciting world of fiber and flax seeds. Probably not but I think this is a super interesting and important subject. A lot of the time in a bid to increase your fiber intake people reach for that box of bran assuming that it is the best and healthiest option. Truth is bran is really rough on your insides. Sure it does the job and gets things moving and keeps things regular, but it almost does too good of a job and really ends up doing a number to your intestinal walls.
So if bran isn't the best option where does that leave you? Well there are actually a massive number of different foods that will help you hit your daily fiber requirements and will help to keep you "regular" as they say but for the sake of today's post I have chosen flax seeds because they are more than just fiber.
Flax seeds are all over the place these days they come in a variety of different forms: whole seeds, ground seeds and oil. The versatility of this seed makes it super easy to incorporate into your regular every day meal plans without too much effort and will hopefully only really provide you with some happy healthy benefits. I say hopefully because everyone is different and people can be allergic or react poorly to anything, so test it out for yourself and see how you're feeling.
So why is flax so great?
In it's seed form the body can't really digest it so it helps to clean out your digestive track and promotes healthy movement of food particles and whatever else is left over in there, helping to keep you regular or get things moving where they once were stalled.
It is full of Omega 3 Fatty Acids. Omega 3 is one of the essential fatty acids that we just don't get enough of in our diets. Sure they have things like omega 3 enriched eggs, milk and whatever other products they have out there, but really the best way to get omega is from the source itself. So foods like flax, fish, and supplements are often the best route. Increasing your Omega 3 levels will not only help on the keeping your regular side of things, it will help in the body's metabolism of fat and can aid in providing some relief due to inflammation. As a runner I can vouch for the latter, I notice a substantial difference in the state of my knees when I increase my Omega 3 intake, I do this by adding flax to my smoothies, salads, granola. In order to get the Omega 3 benefits from flax seed it is best not to cook it, cooking flax denatures the fats.
Flax contains lignans which are linked to helping reduce the risk of breast cancer in women and prostate cancer in men. These same lignans might also help relieve some symptoms of menopause, such as hot flashes.
Flax oil is a great alternative to olive oil or nut oils when making your own salad dressing. It might not be full of fiber but it is full of Omega 3s and has it's own unique nutty flavour, it makes a great addition to your cold oil choices.
Flax is a soluble fiber so it has the chance of helping to lower your cholesterol and risk of heart disease.
All of this being said it is important to note that not everything works for anyone. If you are using flax seed to increase Omega 3 it is important to start off slow and gradually increase the amount you are consuming, because remember it is full of fiber and will therefore act almost laxative like in larger doses. If the goal is to "clean" your system out larger amounts might be what you are looking for. Just be sure to take care in finding out what works for you and your body.
It is also important to note that milled or ground flax seeds and flax oil should be kept in the fridge or somewhere cold. Their make up is pretty finicky and can easily become denatured, which means it might be better to buy whole flax seed and grind them up on your own. But once again play around with what works for you, if pre-ground seeds is what works for you just be sure to store them in the fridge and be aware of their expiry date.
So that is a brief round up of the glories of the all mighty mini flax seed. If you have any other foods you are interested in learning a bit more about let me know and I would be more than happy to write up a post.
This entry was posted in General and tagged benefits, cancer, clean eating, constipation, digestive. care, flax, flax seed, health, menopause by healingginger. Bookmark the permalink.


On 8 June, to mark the first anniversary of the establishment of the European Roma Institute for Arts and Culture (ERIAC), the Berlin-based organization has announced the first seven Barvalipe Academy members nominated by ERIAC's Pakiv Board. These outstanding Romani personalities from arts and the humanities are expected to guide ERIAC in defining its strategy and future activities.
Jarmila Balážová of the Czech Republic is one of the seven figures so honored. Born in Brno in 1972, she is an activist, journalist and moderator who graduated in journalism in 1997 from Charles University.
From 1992-1998 Balážová worked at the public broadcaster Czech Radio for the "O Roma vakeren" program. In 1997 she became the Editor-in-Chief of the Romani monthly Amaro gendalos.
In 1999 she began working for the public broadcaster Czech Television as a dramaturg of programs for children and youth and hosted the children's discussion program "Tykadlo" as well as her own talk show. She was a moderator on the "Áčko" program on the commercial TV NOVA station that same year.
Between 2002 and 2013, she worked with Czech Radio 6 on the programs "Human Rights Focus", "Studio STOP", "Interview", "Focus on Youth" and "Science Talk". From 2003-2014 she was editor-in-chief of the monthly Romano voďi, published by the ROMEA organization.
From its founding until 2014, Balážová was chair of the board of directors of ROMEA. In 2014, she began working as Press Secretary for Czech Human Rights Minister Jiří Dienstbier and since November 2015 she has been Press Secretary for the Czech Education Ministry.
Mihaela Zatreanu, educator and developer of the first curriculum for instruction in the Romanes language in Romania.
ERIAC is also announcing its call for Associate Members, which will close on 31 July. The Institute is a membership organization bringing together Roma and non-Roma individuals and organizations with competencies and experience in arts and culture who want to be part of ERIAC's creative hub.
Associate Members play key roles in building up the resources of ERIAC, engaging in creative exchange and co-production, taking ownership of ERIAC's initiatives, and participating in its decision-making. They have full access to ERIAC's information and knowledge resources, collections, archives and database; are involved in ERIAC's program of activities; are able to shape ERIAC's priorities and activities through the thematic sections; and form part of ERIAC's growing creative community across Europe.
Associate Members are expected to pledge to respect ERIAC's founding principles and to fulfill the principle of contribution. You can apply to join ERIAC by filling out an application form here.


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Renewed Scottish campaign to leave post-Brexit UK
by Deborah Jones | Comments Off
In happier days, Scotland's First Minister, Nicola Sturgeon (R), greeted Britain's new Prime Minister, Theresa May, as she arrived at Bute House in Edinburgh, Scotland, Britain, July 15, 2016. REUTERS/Russell Cheyne
In these times of seething rage, it is increasingly likely that Britain's divorce from the European Union will lead to the break-up of the United Kingdom itself.
As the parliament in Westminster completes the process of giving Prime Minister, Theresa May, authority to start the process of taking Britain out of the European Union, anger and resentment is intensifying in Scotland and in Northern Ireland.
Voters in both Scotland and Northern Ireland opted decisively to remain in the EU in last June's referendum, but have been overruled by the dominant population of England. The unhappiness has been compounded as it has become clear that May is not going to try to fiddle some form of associate status with the EU and is heading for a complete separation, known to headline writers as a "hard Brexit."
These glimpses of the road ahead have prompted the First Minister of Scotland's regional government, Nicola Sturgeon, to say it is "highly likely" she will seek another referendum on Scottish independence. In the first referendum in 2014, Scottish voters decided by a narrow margin to remain in the United Kingdom, not least because that seemed at the time the best option for Scotland to stay in the EU. That equation collapsed when the overriding majority of disenchanted blue collar English voters chose Brexit in last June's referendum.
The situation is particularly fraught in Northern Ireland, where Brexit could wreck the 1998 Good Friday Agreement, aimed at ending over 30 years of sectarian violence between British loyalist Protestants and Irish republican Catholics. An essential element in the confidence-building of the 1998 pact is the open border between Ulster and the Irish republic to the south under the umbrella of the EU.
The re-imposition of a "hard" border between Northern Ireland and Eire will affect progress in demolishing community barriers. Northern Ireland's economy will also be far more adversely affected by Brexit than other parts of Britain because of its dependence on EU agricultural subsidies. And there will be domestic political pressure on the Dublin government to interfere in Northern Irish affairs, thus likely inflaming the passions of Protestant Ulster loyalists.
The peace agreement is already tottering, with the collapse on January 16 of the province's power-sharing executive joining the loyalist Democratic Unionist Party and the republican Sinn Fein. New elections were held on Friday, March 3, but early returns suggest there will be no clear result, and London may have to impose direct rule. That will set back the development of devolved self-government in Northern Ireland. Prolonged direct rule by London could well see the province stripped of the moderate political actors who have begun to mature during the two decades of peace.
Unlike in Scotland, however, there is no clear path by which Northern Ireland could maintain its desired relationship with the EU by, for example, opting to leave the UK and join the Irish Republic.
Sturgeon will set out her argument for a new Scottish independence referendum when she addresses her party's annual meeting later this month. From what she has said and written already, it is evident Sturgeon will make at least three arguments.
One is that her Scottish National Party was elected to the provincial government in May last year (before the Brexit vote) on the understanding that it "should have the right to hold another referendum if there is a significant and material change in the circumstances that prevailed in 2014, such as Scotland being taken out of the EU against our will."
It is unarguable that the June Brexit referendum result was just such a material change in circumstance.
Another argument is that Sturgeon claims Prime Minister May has not fulfilled her promise to only start the process of taking Britain out of the EU once a common approach had been agreed with the British regional governments in Scotland, Wales and Northern Ireland.
Also, her party's voter mandate includes a clear responsibility to keep Scotland within the EU, Sturgeon has said. "There are various ways in which Scotland's place in the European Single Market could be maintained," she wrote in a recent discussion paper.
"One option – in my view, the best option – is to become a full member of the EU as an independent country."
It will be difficult for Theresa May to refuse to give the Scottish Parliament permission to hold another independence referendum. But the British Prime Minister is likely to insist that no referendum be held before the UK has left the EU. May doesn't want to be campaigning in Scotland while she is battling out the future with Brussels.
The negotiations to take Britain out of the EU after 45 years as a member will take at least two years. That means no new referendum on Scottish independence will be held before 2019, and probably later than that.
The European world may well be a different place by that time. With anti-Brussels parties gaining significant influence throughout the EU, and conceivably taking power in Holland, France and Italy this year, the union could be a very different beast, and perhaps even a dead one, by the time the Scots get to vote again on independence.
Even if a new referendum were held now, it is touch-and-go whether Scots would opt to leave the UK. While 60 per cent of Scots opted to stay in the EU in the Brexit referendum, polls suggest they are not as committed to leaving the United Kingdom in order to stick with Europe. Recent public opinion surveys suggest there has been little change in public attitude since 2014, when 55 per cent of Scots decided to stay in the UK.
The economic arguments for Scotland to remain in the UK are compelling. But as we know from the Brexit vote itself, and tribal nationalist convulsions that are consuming the United States, Holland, France and elsewhere, measured economic judgements are not always voters' main consideration.
The Scottish economy is in the doldrums, largely because of the slump in value of its North Sea oil reserves. And EU membership is not as great a direct economic benefit to Scotland as it is for North Ireland. As Scottish writer Alex Massie pointed out recently: "Membership of the UK single market is worth four times as much to Scotland as membership in the EU single market. In these circumstances only a fool would endorse independence."
First Minister Sturgeon counters this argument by warning that Scotland faces tough economic times anyway, that the fall-out from Brexit will be as damaging to Scotland as the necessary austerities of the first years of independence.
"The case for full self-government ultimately transcends the issues of Brexit, of oil, of national wealth and balance sheets and of passing political fads and trends," Sturgeon said recently.
And in this age of demagogy and resurgent tribal nationalism who is to say she is wrong?
Copyright Jonathan Manthorpe 2017
Contact, including queries about syndication/republishing: jonathan.manthorpe@gmail.com
Related from F&O archives:
Scotland Decided: what the experts said in 2014
In its independence referendum, Scotland voted to remain in the United Kingdom by 55 versus 45 per cent. An expert panel looks at what happened, and where it leaves the UK and Scotland.
Scotland's independence referendum: a beginner's guide, September, 2014
Britain's tortuous road to "hard" Brexit, by Jonathan Manthorpe
It is becoming clearer just how wrenching a process it will be for Britain to leave the European Union, and beyond doubt that Britain is headed for a "hard" Brexit.
Brexit will save the European project, by Jonathan Manthorpe
When the dust of history settles, the moment angry Britons voted to quit the European Union will stand out as the moment that saved the 28-nation project.
In England's Mean and Truculent Land, by Jonathan Manthorpe
Britain's departure from the EU will be a journey across new territory full of terrors and treacherous terrain. Among the many stupidities in Cameron's management of the referendum was allowing a simple majority for victory.
Jonathan Manthorpe is a founding columnist with Facts and Opinions and is the author of the journal's International Affairs column. He is the author of "Forbidden Nation: A History of Taiwan," and has been a foreign correspondent and international affairs columnist for nearly 40 years. Manthorpe's nomadic career began in the late 1970s as European Bureau Chief for The Toronto Star, the job that took Ernest Hemingway to Europe in the 1920s. In the mid-1980s Manthorpe became European Correspondent for Southam News. In the following years Manthorpe was sent by Southam News, the internal news agency for Canada's largest group of metropolitan daily newspapers, to be the correspondent in Africa and then Asia. Between postings Manthorpe spent a few years based in Ottawa focusing on intelligence and military affairs, and the United Nations. Since 1998 Manthorpe has been based in Vancouver, but has travelled frequently on assignment to Asia, Europe and Latin America.
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Posted in Also tagged Brexit, European Union, Nicola Sturgeon, Scottish independence |
Analysis: China's belligerence, Southeast Asia's arms race, and 1914
International affairs columnist Jonathan Manthorpe examines the echoes of the first World War in the arms race underway in Southeast Asia. Excerpt:
As China has become wealthy it has also become an expansionist power. Beijing portrays its territorial ambitions as merely the re-assertion of its sovereignty over lands that were stolen from it during the "century of humiliation" after its disastrous collision with the industrial powers starting in the 1840s.
For the most part these claims are bunk. Beijing's loud and often belligerent claims to ownership of the Japanese-controlled Senkaku Islands in the East China Sea, the island nation of Taiwan, and most of the South China Sea as far as the territorial waters of Indonesia are modern fabrications of little or no merit.
At the same time, Chinese officials have studied the history of the European empires of the 18th and 19th centuries and concluded that their survival and expansion depended on the control of far-flung resources. Because of its communist and authoritarian heritage and instincts, Beijing continues to shy away from the message of the American imperium and to trust the market place to provide what is necessary to fuel its economy.
To these ends to control of territory and resources, China has in the last 20 years been pursuing the ability to project power to defend its interests. The key ingredient of this effort has been the construction of a massive and modern blue water navy, whose evident purpose is to challenge the supremacy of the U.S. navy.
It is this that carries echoes of the years leading up to 1914 and the start of the First World War.
Log in to read the column, 1914 and the China Syndrome.* (F&O premium works, including our commentary, are available for a $1 site day pass, or with monthly or annual subscriptions. Real journalism has value, and to avoid the conflicts inherent in advertising or soliciting outside funding F&O relies entirely on reader payments to sustain our professional quality.)
Posted in All, Current Affairs Also tagged Chinese militarization, Diaoyu, East China Sea, imperialism, Japan, Senkaku Islands, South China Sea, Southeast Asian arms race, Taiwan, World War One |
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