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

It is by now an often repeated fact that the  so-called
(single-band) Hubbard model
was originally introduced to explain ferromagnetism
\cite{gutzw63,hubba63,kanam63}. In what followed, however, it turned
out to be rather a generic model for antiferromagnetism.
Ferromagnetism seemed to require additional ingredients, for instance
the existence of degenerate bands which favor ferromagnetism based on
Hund's rule or in the insulating case certain additional ferromagnetic
couplings and/or correlated hopping terms. Both scenarios were proven
 rigorously
in recent years (see  \cite{mielk93b} and references therein for the former
 and \cite{kolla96} and references therein for the latter).

The  Hubbard model and its possible ferromagnetic ground
state are of renewed interest \cite{fazek97,vollh97}.
 There are many works in the
field based on quasi one dimensional  ($d=1$) systems triggered by the
prediction of ferromagnetism in double minima systems at
low particle density \cite{mulle95} and by the numerous possibilities
of analytical and numerical calculations
\cite{penc96a,pieri96,daul96,sreer97} in $d=1$.
Exact calculations are possible
in infinite dimensions ($d=\infty$) \cite{uhrig96a,ulmke96}. For
intermediate dimensions ($1< d < \infty$) numerical and approximate methods
are employed \cite{hlubi97,herrm97a,herrm97b}.

An important mile stone in the research of ferromagnetism in Hubbard
models is the work of Nagaoka \cite{nagao65,tasak89}. It showed that at
infinite local repulsion a single electron above half-filling favors
the saturated ferromagnetic ground state (henceforth: Nagaoka state)
if the underlying lattice has loops which allow interference.
For bipartite lattices particle-hole symmetry extends these
results to hole doping. This result reveals the beauty and the difficulty
of the question for which lattices and for which fillings the Nagaoka
state is the ground state. At $U=\infty, T=0$ there is only the hopping
left as a global energy scale. Thus there is no expansion parameter,
 no adiabatic limit, and
no competition of energy scales. The issue is solely a question of
the lattice structure, i.e.\ the possible paths on the lattice, and of
the filling.

Unfortunately, there are no extensions of Nagaoka's result to
macroscopic dopings. Only non-macroscopic numbers of holes could be treated
\cite{barbi90,tian91}. Therefore, we choose another route in the present
work and investigate the stability of the Nagaoka state towards a single
spin flip. If such a flip lowers the energy then the Nagaoka state is not
the ground state. Otherwise it is locally stable. The drawback that we
treat only local stability in this way is not very serious. There is no
indication that the transition away from saturation should not be
continuous at $T=0$, see e.g. \cite{takah69}.

A more serious drawback is the fact that even the single spin flip
is too difficult a problem to be solved completely on finite dimensional
lattices. In the limit of infinite dimensional lattices, however,
it was solved \cite{uhrig96a}. Thereby it was shown that
relatively simple variational ansatzes provide already a qualitative
insight in the tendency of a certain lattice to have a Nagaoka state
as ground state. Wurth {\em et al}. showed that only
an extremely sophisticated variational ansatz \cite{wurth96}
yields a further reduction
of the region of possible Nagaoka state stability
in comparison to simpler ansatzes \cite{hanis93}.

It is the aim of the present paper to extend previous work
on variational ansatzes decisively \cite{hanis95}, both in the
completeness of the ansatzes and in the types of the lattices considered.
So far, variational ansatzes considered a finite vicinity of the flipped
spin and treated a finite number of parameters leading to matrix
eigenvalue problems. Here we will show that a resolvent approach is
capable to deal implicitly with an \textit{infinite} number of
variational parameters. No explicit knowledge of the variational
wave function is required. A similar approach was used recently
by Okabe\cite{okabe97} for the square lattice and the
simple cubic lattice, too. In his work,
however, the reduction of the resolvent to simple integrals over the
density of states (DOS),
which we succeeded to achieve in most cases, is lacking.

We will present elegant simple expressions for the
Nagaoka instability line
$U_{\rm cr}(n)$ which apply to most common lattices. These results
make it possible for everyone to check easily whether or not one can
expect a ferromagnetic ground state for a given lattice.
We will show that two main features favor the occurrence of
a saturated ferromagnetic ground state
\begin{enumerate}
\item
a highly asymmetric density of states with large values at the
lower band edge (after particle-hole transformation).
\item
non-bipartiteness of the lattice, i.e.\ frustration due to loops of
three sites.
\end{enumerate}
Of course, the two points are intimately related.

The setup of our article is as follows. In the rest of the Introduction
we will present certain variational ansatzes used so far to investigate
the Nagaoka state stability. In the subsequent section \ref{sect:resapp}
we develop the resolvent approach which yields simple formulae
for the stability lines on homogeneous, isotropic lattices
with nearest neighbor hopping. In sect.\ \ref{sect:resvarlat}
we present our
results for various lattices in dimensions $d=2$ and $d=3$,
namely the square, the simple cubic, the \textit{bcc}, the honeycomb,
the triangular, the kagome, and the \textit{fcc} (\textit{hcp}) lattice. For
the $t$-$t'$ Hubbard model on the square lattice a perturbative approach
for small $|t|$ is employed as well. Sect.\
\ref{sect:conclus} contains
a summary and a final discussion of the lattice dependence of saturated
ferromagnetism in the Hubbard model.
The appendices contain technical details in the derivation for the various
lattices.

\subsection{Preliminary approaches}

We consider the conventional single band Hubbard model
\begin{equation} \label{def:model}
H = -t \sum_{\langle \underline{i}, \underline{j}\rangle \sigma}
a_{\underline{i} \sigma}^{+} a_{\underline{j} \sigma} + U
\sum_{\underline{i}} a_{\underline{i} \uparrow}^{+}
a_{\underline{i} \uparrow} a_{\underline{i} \downarrow}^{+}
a_{\underline{i} \downarrow}
\end{equation}
and calculate the spin flip energy
\begin{equation} \label{def:esf}
\Delta e = \langle \Psi | H - E_{\cal N} | \Psi \rangle / \langle \Psi |
\Psi \rangle
\end{equation}
where $E_{\cal N}$ is the energy of the Nagaoka state and $|\Psi \rangle$ is
a variational wave function.  Whenever $\Delta e < 0$
the Nagaoka state is definitely not the ground state due to the variational
nature of our approach. At $U = \infty$, the zero of $\Delta e_{\infty}
(\delta) := \Delta e (U=\infty, \delta)$ gives the critical hole density
$\delta_{\rm cr}$ above which the Nagaoka state is unstable. For finite $U$,
$\Delta e (U, \delta) \doteq 0$ leads to the Nagaoka instability
line $U_{\rm cr} (\delta)$
which separates a region of guaranteed instability of the Nagaoka state
($U<U_{\rm cr}(\delta)$) in the phase diagram of the Hubbard model
from a region of possible stability of the Nagaoka state
($U>U_{\rm cr}(\delta)$).
In the phase diagrams displayed in this paper we will always represent the
on-site repulsion $U$ in terms of $U_{\rm red} = U/(U+U_{\rm BR})$ where
$U_{\rm BR} = -16 \epsilon^{0}$ denotes the Brinkman-Rice critical
coupling\cite{brink70}. $\epsilon^{0}$ is
the energy per particle of the saturated ferromagnetic state for the
quarter-filled band and depends on the underlying lattice.
This representation is chosen to render comparisons between different
lattices possible.

A common starting point\cite{shast90a,mulle91,mulle93b,hanis93,hanis95}
 is defined by the ansatz
\begin{eqnarray} \nonumber
| \Psi \rangle &:=& |\Lambda|^{-1/2} \sum_{\underline{i}}
\exp( i \underline{k}_{\rm b} \underline{i}) \,
\big[a_{\underline{i} \uparrow} \big(a_{\underline{i} \uparrow}^{+} +
f \cdot \sum_{\langle \underline{i}, \underline{j}\rangle}
a_{\underline{j} \uparrow}^{+}\big)
\\ \label{ansatz:gw}
&+&
g \cdot a_{\underline{i} \uparrow}^{+} a_{\underline{i} \uparrow}\big]
a_{\underline{i} \downarrow}^{+} | {\cal N}' \rangle \ .
\end{eqnarray}
For $f=0$  this is the Gutzwiller single spin flip wave function (Gw). The
parameter $g$ controls the probability of double occupancy.
The system size is denoted by $|\Lambda|$.
We use the operators $a \, (a^{+})$ for site diagonal fermion
annihilation (creation)
and $c \, (c^{+})$ for momentum diagonal fermion annihilation (creation).
Furthermore, we use $n$ for the particle density, $\delta = 1-n$ for
the doping per site, $z$ for the coordination number, and
$e_1=E_{\cal N}/|\Lambda|$
 for the expectation value of the kinetic energy.
The ket $| {\cal N}' \rangle
= c_{\underline{k}_{\rm F} \uparrow} | {\cal N} \rangle$ is the fully
polarized Fermi sea of $\uparrow$-electrons from which one
${\rm e}_{\uparrow}^{-}$ at the Fermi level $\varepsilon_{\rm F}$
is removed.
The energy balance of (\ref{ansatz:gw}) with $f=0$
reads at infinite $U$ ($g=0$) (see (\cite{shast90a,hanis93})
\begin{equation}
\label{skaband}
\Delta e = -e_1/\delta -\varepsilon_{\rm F} +
\varepsilon_{\underline{k}}\delta(1-(e_1/\delta z t)^2)
\end{equation}
where $\varepsilon_{\underline{k}}$ is the dispersion.
The maximum energy lowering is obviously obtained for $\underline{k}$
 belonging
to the lower band egde $\varepsilon_{\rm b}$, i.e.\ here
$\underline{k}_{\rm b}=\underline{0}$.

For finite $f$
 majority spin hopping processes from the position of the flipped spin
to nearest neighbor sites are taken into account. This ansatz will be
denoted NN. The amplitudes of these hopping
processes are assumed to reflect the lattice symmetry.
Basile and Elser investigated an ansatz similar to NN which
includes $\uparrow$-hopping processes from the position of the
$\downarrow$-electron to \textit{all} other lattice sites \cite{basil90}.
Since the number of variational parameters increases with the lattice size
they only studied a finite square lattice. The resolvent method developed
in sect.\ \ref{sect:resapp} allows us to investigate a variational ansatz
equivalent to the full Basile-Elser wave function \textit{in the
thermodynamic limit} on all common lattices. We also derive
improved variational criteria for the Nagaoka instability at $U<\infty$ by
 extending the Hilbert subspace further.

\section{Resolvent approach} \label{sect:resapp}
Generally, a resolvent is an operator-valued expression
of the type
a\begin{equation}
\label{resallg}
R(\omega) = 1/(\omega-(H-E_{\cal N}))
\end{equation}
 where $H-E_{\cal N}$ is the
Hamilton operator with respect to the ground state
energy $E_{\cal N}$ (here: the Nagaoka state energy).
 From (\ref{resallg}) it is clear that the existence of
any state at $\omega=\omega_0$ implies a pole or at least
a singularity in the resolvent. For this reason, we will
investigate in the following the resolvent $R$ applied to
$c^+_{\underline{k}_{\rm b}\downarrow}|{\cal N}'\rangle$
and compare $\omega_0$ to $\varepsilon_{\rm F}$.

It is not possible to compute $R$ for the whole
Hilbert space except under simplifying conditions like
infinite coordination number \cite{uhrig96c}. Hence we
will restrict the inversion to certain subspaces which still allow an
analytical treatment. The results obtained in this way
for the lower band edge are variational.
This means that excitation energies found
 are upper bounds to the true ones
and that specific interaction values $U$ come
out too small compared to those of the full solution.

\subsection{Case $U=\infty$: Ansatz RES0}
For infinite on-site repulsion no double occupancy is allowed.
Thus at the site of the $\downarrow$-e$^-$ no $\uparrow$-e$^-$
is allowed. We investigate therefore the variational subspace
spanned by
$a_{\underline{i}\uparrow}^{\phantom{+}} a^+_{\underline{j}\uparrow}
a^+_{\underline{i}\downarrow}|{\cal N}'\rangle$ with
arbitrary $\underline{i}$ and $\underline{j}$.
We define
\begin{mathletters}
\label{varia}
\begin{equation}
| \Phi_{\underline{k}} \rangle
 := {\cal A}_{\underline{k}} | {\cal N}' \rangle
\end{equation}
\begin{equation}
{\cal A}_{\underline{k}}  :=
|\Lambda|^{-1/2}\sum\limits_{\underline{i}}
 \exp(i(\underline{k}_{\rm b}- \underline{k})\underline{i})
a_{\underline{i}\uparrow}^{\phantom{+}}
c_{\underline{k}\uparrow}^{+}
a_{\underline{i}\downarrow}^{+}  \ ,
\end{equation}
\end{mathletters}
where the admissible values of $\underline{k}$ are
outside the Fermi sphere (FS), but inside the Brillouin zone (BZ),
i.e.\ $\underline{k} \in {\rm BZ} \setminus {\rm FS}$.
The Hamiltonian does not mix states (\ref{varia}) for
{\em different} total momenta $\underline{k}_{\rm b}$.
States (\ref{varia}) for
{\em different} total momenta $\underline{k}_{\rm b}$ are orthogonal.
Ansatz (\ref{varia}) contains in particular the NN ansatz (\ref{ansatz:gw})
and of course the simple Gutzwiller ansatz. It comprises
$\uparrow$-hopping processes of arbitrary distance,
i.e.\ it is the thermodynamic extension
of the ansatz investigated previously by Basile and Elser \cite{basil90}.

For the computation of the resolvent $R(\omega)$ one can use
the Mori/Zwanzig projection formalism
(see e.g.\ appendix C in \cite{fulde93}) with the scalar product
$({\cal A} |{\cal B}) := \langle {\cal N}' |
[{\cal A}^+,{\cal B}]_+ | {\cal N}' \rangle$ for the operators
${\cal A}$ and ${\cal B}$.
The resolvent (\ref{resallg}) then becomes
\begin{eqnarray}\nonumber
{\bf R}_{\underline{k}_{1}, \underline{k}_{2}} (\omega) &=&
\langle \Phi_{\underline{k}_{1}}| R(\omega)
|\Phi_{\underline{k}_{2}} \rangle
\\ \label{resmori}
&=&
({\mathcal{A}}_{\underline{k}_{1}} |
(\omega - {\cal L})^{-1} {\mathcal{A}}_{\underline{k}_{2}})\ .
\end{eqnarray}
Here the Liouville operator ${\cal L}$ is used which is defined as
${\cal LA}:=[H,{\cal A}]$ for all operators ${\cal A}$ \cite{fulde93}.
The resolvent can be expressed in matrix notation \cite{fulde93} by
\begin{equation}
\label{resolv}
{\bf R}(\omega) = {\bf P} \, (\omega {\bf P} -
{\bf L} - {\bf M}(\omega))^{-1}
\, {\bf P}
\end{equation}
with the norm matrix ${\bf P}$ and the frequency matrix ${\bf L}$
\begin{mathletters}
\begin{equation}
{\bf P}_{\underline{k}_{1}, \underline{k}_{2}} :=
\langle \Phi_{\underline{k}_{1}} | \Phi_{\underline{k}_{2}} \rangle
\end{equation}
\begin{equation}
{\bf L}_{\underline{k}_{1}, \underline{k}_{2}} :=
\langle \Phi_{\underline{k}_{1}} | H -
E_{\cal N} | \Phi_{\underline{k}_{2}} \rangle \ .
\end{equation}
\end{mathletters}
The frequency matrix ${\bf L}$ encodes the effect of $H$ in the subspace
considered. The deviation of  ${\bf P}$ from unity accounts for the
non-orthonormality of the basis. The so-called memory matrix
${\bf M}(\omega)$ describes the effect of all processes which
imply excursions outside the subspace considered. If the ground
state is known exactly (which holds in the present case) the
approximation ${\bf M}(\omega)={\bf 0}$ is variational in nature
for the lower band edge.

It is the aim of the subsequent calculation to obtain a simple
condition for the singularity of $(\omega {\bf P} - {\bf L})$.
This singularity then signals that $\omega$ corresponds to
an eigen energy. To this end, we first need the matrix elements
\begin{mathletters}
\label{xelem}
\begin{equation}
\label{xelema}
{\bf P}_{\underline{k}_{1}, \underline{k}_{2}} =
n \, \delta_{\underline{k}_{1}, \underline{k}_{2}} + |\Lambda|^{-1}
\end{equation}
\begin{equation}
\label{xelemb}
{\bf L}_{\underline{k}_{1}, \underline{k}_{2} \uparrow} =
\delta_{\underline{k}_{1}, \underline{k}_{2}} (n \cdot
\varepsilon_{\underline{k}_{2}} - e_1)
\end{equation}
\begin{equation}
\label{xelemc}
{\bf L}_{\underline{k}_{1}, \underline{k}_{2} \downarrow}
= |\Lambda|^{-1}
 \, \varepsilon_{\rm b} - \delta_{\underline{k}_{1},
\underline{k}_{2}} (zt)^{-1} e_1
\, \varepsilon_{\underline{k}_{2} - \underline{k}_{\rm b}} \
\end{equation}
\end{mathletters}
 We use the notation  $e_i:= \langle
\Theta(\varepsilon_{\rm F}-\varepsilon_{\underline k})
\varepsilon_{\underline k}^i \rangle_{\rm BZ}$ ($\Theta(\varepsilon)$
is the Heaviside function).
The elements in (\ref{xelem})
 are obtained with the help of Wick's theorem since
$|{\cal N}'\rangle$ is a simple Slater determinant.
In (\ref{xelemb}) and (\ref{xelemc}), we distinguish the part
coming from the motion of the $\uparrow$-electrons and the part
coming from the motion of the $\downarrow$-electron.
The expression $(zt)^{-1} e_1
\, \varepsilon_{\underline{k}_{2} - \underline{k}_{\rm b}}$
in (\ref{xelemc}) is obtained from
\begin{eqnarray}\nonumber
&&-(2\pi)^{-d}\int\limits_{\underline{k}_1 \in {\rm BZ} \setminus {\rm FS}}
\varepsilon_{\underline{k}_2-\underline{k}_{\rm b}- \underline{k}_1}
d^dk_1 =
\\ \label{isotrop}
&& \qquad \frac{1}{zt}\varepsilon_{\underline{k}_2-\underline{k}_{\rm b}}
\underbrace{
(2\pi)^{-d}\int\limits_{\underline{k}_1 \in {\rm BZ} \setminus {\rm FS}}
\varepsilon_{\underline{k}_1} d^dk_1}_{e_1:=} \ .
\end{eqnarray}
This relation
holds for all homogeneous, isotropic lattices with NN hopping only, e.g.\
square lattice, triangular lattice, kagome lattice and so on.
The result  (\ref{isotrop}) can be found easiest by interpreting
the left hand side as convolution of
$\varepsilon_{\underline{k}_2-\underline{k}_{\rm b}}$ and of
$\Theta(\varepsilon_{\rm F}-\varepsilon_{\underline{k}_1})$ ,
i.e.\ as a  multiplication in real space which concerns only the NN terms.
Thus it is the multiplication with a constant
$\langle a_{\underline{j}}^{\phantom{+}} a_{\underline{i}}^+\rangle
= -e_1/(zt)$. The sites $\underline{i}$ and
$\underline{j}$ are arbitrary adjacent sites since all
 bonds are equal due to the
required homogeneity and spatial isotropy.

On the basis of (\ref{xelem}) the matrix inversion can be rephrased as
\begin{equation}
\label{invert}
(\omega {\bf P} - {\bf L})^{-1} =
\left({\bf d}^{-1} + (\omega - \varepsilon_{\rm b}) \,
{\underline v} \, {\underline v}^{+}\right)^{-1} \
\end{equation}
with the constant vector ${\underline v}=|\Lambda|^{-1/2}$ and the
diagonal matrix ${\bf d}_{{\underline k}_{1} {\underline k}_{2}} =
\delta_{{\underline k}_{1} {\underline k}_{2}}  f({\underline k}_2)$ with
\begin{equation}
\label{fdef}
f(\underline k) := [n(\omega -\varepsilon_{\underline k}) +
e_1 (1 + (zt)^{-1}
\varepsilon_{\underline k -\underline{k}_{\rm b}})]^{-1} \ .
\end{equation}
Note that the dyadic product ${\underline v} \, {\underline v}^{+}$
provides a $\delta|\Lambda| \times \delta|\Lambda|$ matrix
with the constant matrix element $|\Lambda|^{-1}$.

Expanding the right hand side of (\ref{invert}) in terms
of ${\underline v} \, {\underline v}^{+}$ and resummation in terms of
\begin{mathletters}
\begin{eqnarray}
h(\omega)&:=&  {\underline v}^{+} {\bf d}\, {\underline v}\nonumber \\
& =&  (2\pi)^{-d} \int\limits_{\underline k \in {\rm BZ}
\setminus {\rm FS}}
f(\underline k) d^dk \nonumber \\
&=&  \langle \Theta(\varepsilon_{\underline k} - \varepsilon_{\rm F}) \,
f(\underline k) \rangle_{\rm BZ}
\label{hdef0}
\end{eqnarray}
 yields
\begin{eqnarray}
&&(\omega {\bf P} - {\bf L})^{-1} \nonumber \\
 && = ({\bf d}^{-1} + (\omega - \varepsilon_{\rm b})
{\underline v} {\underline v}^{+} )^{-1} \nonumber \\
 && = {\bf d} \, ( {\bf 1} +
(\omega - \varepsilon_{\rm b}) {\underline v} {\underline v}^{+}
{\bf d})^{-1} \nonumber \\
 && = {\bf d} \, ( {\bf 1} -
(\omega - \varepsilon_{\rm b}){\underline v}{\underline v}^{+} {\bf d}
 + (\omega - \varepsilon_{\rm b})^{2} {\underline v}{\underline v}^{+}
{\bf d} {\underline v} {\underline v}^{+} {\bf d}
 + \ldots) \nonumber \\
 && = {\bf d} - (\omega - \varepsilon_{\rm b}) \,
{\bf d} {\underline v} {\underline v}^{+} {\bf d}
 \, (1 + (\omega - \varepsilon_{\rm b}) \, h(\omega))^{-1} \ .
\label{funda0}
\end{eqnarray}
The matrix elements thus read
\begin{eqnarray} \nonumber
&&\left((\omega{\bf P}-{\bf L})^{-1}\right)_{{\underline k}_{1},
{\underline k}_{2}} =
\delta_{{\underline k}_1, {\underline k}_2} \, f({\underline k}_1)
\\
&& \qquad
- \frac{\omega - \varepsilon_{\rm b}}{1+ (\omega - \varepsilon_{\rm b})
h(\omega)} \cdot
\frac{f({\underline k}_1) \, f({\underline k}_2)}{|\Lambda|} \ .
\label{funda1}
\end{eqnarray}
\end{mathletters}
From (\ref{funda1}) we read off that $(\omega {\bf P} - {\bf L})$
is singular for
\begin{equation}
\label{res0}
0=1 + (\omega - \varepsilon_{\rm b}) \, h(\omega)\ .
\end{equation}
The trick to reduce dyadic perturbations to simple divisions is
commonly known under the name `Householder method'
in the numerics of matrices.
This extremely simple result is derived here for all Bravais lattices,
e.g.\ the square lattice, the triangular lattice, but not
for the honeycomb lattice or the kagome lattice.
The restriction to Bravais lattices enters since we implicitly
assume that there is one eigen state for each value of ${\underline k}$
in the one-particle Hamiltonian. But it will be shown in
appendix \ref{app:unfrustrated} that identical formulae
apply for general
unfrustrated lattices. Similar formulae can be found for frustrated
non-Bravais lattices, for instance the kagome lattice in
appendix \ref{app:kagome}.

In appendix \ref{app:tri} it is explained that
computing $h(\omega)$ for non-bipartite lattices requires explicit
integration over the momenta.
For lattices  where the band minimum $\varepsilon_{\rm b}$ is reached at
${\underline k}_{\rm b}={\underline 0}$ further
simplification is possible.
The band minimum is found at $\underline{k}_{\rm b}=\underline{0}$
in particular for bipartite lattices
where one may choose $t>0$ without loss of generality.
Then the term $\varepsilon_{\underline k -\underline{k}_{\rm b}}$ in
(\ref{fdef})
reduces to the unshifted dispersion and the whole integration
in (\ref{hdef0}) can be written as integration over the
density of states (DOS) $\rho(\varepsilon)$.
\begin{mathletters}
\label{kb0def1}
\begin{equation}
\label{hdef}
h(\omega) = \int\limits_{\varepsilon_{\rm F}}^{\varepsilon_{\rm t}}
\frac{\rho (\varepsilon) \, d\varepsilon}{n \omega + e_{\cal N}
- \gamma \varepsilon} = \frac{1}{\gamma} \ G(\Omega)
\end{equation}
\begin{eqnarray}
\label{Gdef}
G(y) &:=& \int\limits_{\varepsilon_{\rm F}}^{\varepsilon_{\rm t}}
\frac{\rho(\varepsilon) \, d\varepsilon}{y - \varepsilon} \\
 \gamma &:=& n - e_1/zt \label{gamdef}
\\
\Omega &:=& (n \omega + e_1)/\gamma
\end{eqnarray}
\end{mathletters}

We will call the ansatz deduced from the subspace
given in (\ref{varia}) RES0. It leads to the singularity condition
(\ref{res0}) or to its generalizations for non-Bravais lattices.

Once the energy $\omega$ is found
from (\ref{res0}) for a given Fermi energy $\varepsilon_{\rm F}$
the spin flip energy for the whole process of taking
one $\uparrow$-e$^-$ out at the Fermi level and inserting it as
$\downarrow$-e$^-$ at the lowest possible energy is given by
\begin{equation}
\label{spinflip2}
\Delta e_\infty = \omega -\varepsilon_{\rm F} \ .
\end{equation}
A critical doping $\delta_{\rm cr}$ is found where this spin flip
energy vanishes.

\subsection{Case $U<\infty$: ansatzes RES1, RES2, and RES3}
Besides the calculation of variational upper bounds for spin flip
energies and resulting critical dopings it is our aim to determine
critical interaction values $U$.

For $U<\infty$ we have to include states with double occupancy.  The
easiest way to do so is to include a local double occupancy
\cite{gutzw65,shast90a}. This is done in the ansatz RES1 by adding to
the states defined in (\ref{varia}) the state
\begin{equation}
\label{varib}
| \Psi_1 \rangle
 :=  |\Lambda|^{-1/2}\sum\limits_{\underline{i}}
 \exp(i\underline{k}_{\rm b} \underline{i})
a_{\underline{i}\uparrow}^{+}
a_{\underline{i}\uparrow}^{\phantom{+}}
a_{\underline{i}\downarrow}^{+} | {\cal N}' \rangle \ .
\end{equation}
This ansatz contains the nearest neighbor ansatz NN (\ref{ansatz:gw})
(and the Gutzwiller ansatz) for $U<\infty$.
Again we want to compute the resolvent (\ref{resolv}).
To do so the parts computed in the previous
subsection can be used again. The matrices for RES1
have the block structure
\begin{mathletters}
\label{block1}
\begin{equation}
{\bf P} = \left(
\begin{array}{c|c}
  {\bf P_1} & {\underline 0}^+        \\
  \hline
  {\underline 0}  & P_2
\end{array}
\right) \ \
\omega{\bf P} - {\bf L} = \left(
\begin{array}{c|c}
  {\bf D_1} & {\underline N}        \\
  \hline
  {\underline N}^+   & D_2
\end{array}
\right)
\end{equation}
\begin{equation}
(\omega{\bf P} - {\bf L})^{-1} = \left(
\begin{array}{c|c}
  {\bf B_1} & {\underline M}        \\
  \hline
  {\underline M}^+   & B_2
\end{array}
\right)\ .
\end{equation}
\end{mathletters}
The matrices ${\bf P_1}$ and ${\bf D_1}$ are the same as in (\ref{funda1})
at $U=\infty$. The null vector ${\underline 0}$ in ${\bf P}$ comes from
the fact
that the state with double occupancy
$| \Psi_1 \rangle$ is orthogonal to the states without double
occupancy $| \Phi_{\underline k} \rangle$. The other matrix elements are
again found by Wick's theorem
\begin{mathletters}
\label{xelem2}
\begin{eqnarray}
P_2 &=& n
\\
D_2 &=& n(\omega - U) + e_1 - \varepsilon_{\rm b} \left(n^{2} -
\left( e_1/(zt)
\right)^{2} \right)
\\
N_{\underline k} &=&   -|\Lambda|^{-1/2} \left(
n (\varepsilon_{\rm b} - \varepsilon_{\underline k}) +
e_1 (1 +\varepsilon_{\underline k - \underline{k}_{\rm b}}/(zt)) \right)
\end{eqnarray}
\end{mathletters}
Since we are at present only interested in the singularity condition
it is sufficient to compute one of the elements of
$\omega{\bf P}- {\bf L}$. The easiest is $B_2$, for which an argument
similar to the one leading to (\ref{funda0}), yields
\begin{equation}
B_2 = (D_2 - {\underline N}^+ {\bf D}_{{\bf 1}}^{-1}
{\underline N})^{-1}\ .
\end{equation}
Thus the singularity condition simply reads
\begin{equation}
\label{singcond}
0 \doteq D_2 - {\underline N}^+ {\bf D}_{{\bf 1}}^{-1} {\underline N}
\end{equation}
Now it is advantageous that ${\bf D_1}^{-1}$ is already given in
(\ref{funda1}). Inserting (\ref{xelem2}) one obtains after some
cancellations
\begin{equation}
\label{res1}
\omega - \varepsilon_{\rm b} - n U (1 +
(\omega - \varepsilon_{\rm b})  h(\omega)) \doteq 0 \ .
\end{equation}
Equation (\ref{res1}) is as simple as (\ref{res0}) and enables
us to calculate critical $U$ values explicitly.
Setting $\omega=\varepsilon_{\rm F}$ in (\ref{res1}), which according to
(\ref{spinflip2}) corresponds to vanishing spin flip energy,
renders $U_{\rm cr}$ directly accessible:
\begin{equation}
\label{ures1}
U_{\rm cr}^{\rm RES1} (\delta) = \frac{\varepsilon_{\rm F} -
\varepsilon_{\rm b}}{(1- \delta) \,
[1+ (\varepsilon_{\rm F} - \varepsilon_{\rm b}) \, h(\varepsilon_{\rm F})]} \ .
\end{equation}

It turns out, however, that the values for $U_{\rm cr}$ from
(\ref{ures1}) are not very good close to half-filling $n=1$ where
antiferromagnetic exchange processes are important.  These are not
accounted for in (\ref{varib}). They are considered, at least
to a certain extent, in the ansatz RES2 by using
\begin{equation}
\label{varic}
| \Psi_2 \rangle
 :=  |\Lambda|^{-1/2}\sum\limits_{<\underline{i}\underline{j}>}
 \exp(i\underline{k}_{\rm b} \underline{i})
a_{\underline{i}\uparrow}^{+}
a_{\underline{j}\uparrow}^{\phantom{+}}
a_{\underline{i}\downarrow}^{+} | {\cal N}' \rangle
\end{equation}
instead of $| \Psi_1 \rangle$ as extension of the RES0 subspace.

The block structure (\ref{block1}) remains the same and so does
the singularity condition (\ref{singcond}). Only the matrix elements
are modified
\begin{mathletters}
\label{xelem3}
\begin{eqnarray}
&&P_2 = (e_1^2 + \delta e_2)/t^2
\\
\nonumber
&&D_2= ((e_1^2 + \delta e_2)(\omega - U) +
e_1 e_2 +\delta e_3
\\ \label{d2res2}
&& \qquad - \varepsilon_{\rm b}
\left(e_1^{2} -  e_1e_3/(zt)^2\right))/t^2
\\
&&N_{\underline k} = \nonumber \\
&&   |\Lambda|^{-1/2} \left( e_1
(\varepsilon_{\rm b} - \varepsilon_{\underline k}) +
e_2\left(1 +  \varepsilon_{\underline k -
\underline{k}_{\rm b}}/(zt)\right) \right)/t \ .
\end{eqnarray}
\end{mathletters}
The explicit expression resulting now
 from (\ref{singcond}) is less transparent
than (\ref{res1}) since no cancellations occur. We focus here on the most
important case ${\underline k}_{\rm b}={\underline 0}$.
In addition to the definitions
(\ref{kb0def1}) we use
\begin{mathletters}
\begin{eqnarray}
\gamma' &:=& e_1 - e_2/(zt)
\\
\Omega_{\rm b} &:=& (e_1\varepsilon_{\rm b} + e_2)/\gamma'
\\
\label{ydef}
y &:=& (\gamma'/\gamma)\left(\delta +
(\Omega_{\rm b}-\Omega)G(\Omega)\right)
\end{eqnarray}
\end{mathletters}
and obtain from (\ref{singcond})
\begin{eqnarray}
D_2 &\doteq& \sum\limits_{{\underline k}_1,{\underline k}_2 \in
{\rm BZ}\setminus{\rm FS}}
N_{{\underline k}_1}^+ ({\bf D_1}^{-1})_{{\underline k}_1,
{\underline k}_2}N_{{\underline k}_2}
\nonumber \\ \nonumber
&=& \gamma'[(\Omega_{\rm b} - \Omega) y + \frac{\gamma'}{\gamma}
(\delta\Omega_{\rm b} + e_1)]
\\
&& \quad - y^2
\frac{\omega - \varepsilon_{\rm b}}{1+(\omega -
\varepsilon_{\rm b})h(\omega)}
\label{res2}
\end{eqnarray}
from which $U_{\rm cr}$ can easily be determined. The value
$U_{\rm cr}$ appears only in
$D_2$, see (\ref{d2res2}). The results of RES2 (\ref{res2}) generically
lead to $U_{\rm cr}\propto 1/\delta$ on vanishing doping. In this sense it
represents an important improvement over RES1 (\ref{res1}). For explicit
results we refer the reader to the next section.

At last in RES3, we generalize the variational states with double occupancy
like (\ref{varib}) and (\ref{varic}) in the same manner as we
generalized the states without double occupancy in (\ref{varia})
\begin{mathletters}
\label{varid}
\begin{equation}
| \Psi_{\underline{k}} \rangle
 := {\cal B}_{\underline{k}} | {\cal N}' \rangle
\end{equation}
\begin{equation}
{\cal B}_{\underline{k}}  :=
|\Lambda|^{-1/2}\sum\limits_{\underline{i}}
 \exp(i(\underline{k}_{\rm b} + \underline{k})\underline{i})
a_{\underline{i}\uparrow}^{+}
c_{\underline{k}\uparrow}^{\phantom{+}}
a_{\underline{i}\downarrow}^{+}  \ ,
\end{equation}
\end{mathletters}
where now the admissible values of $\underline{k}$ are all vectors
inside the Fermi sphere (FS).
Note that the extension RES3 contains both RES1 and RES2.
The block structure of the resulting problem is
similar to the one in (\ref{block1}). The difference is that
all blocks are now macroscopically large
\begin{mathletters}
\label{block2}
\begin{equation}
{\bf P} = \left(
\begin{array}{c|c}
  {\bf P_1} & {\bf 0}^+        \\
  \hline
  {\bf 0}   & {\bf P_2}
\end{array}
\right), \quad
\omega{\bf P} - {\bf L} = \left(
\begin{array}{c|c}
  {\bf D_1} & {\bf N}        \\
  \hline
  {\bf N}^+   & {\bf D_2}
\end{array}
\right)
\end{equation}
\begin{equation}
(\omega{\bf P} - {\bf L})^{-1} = \left(
\begin{array}{c|c}
  {\bf B_1} & {\bf M}        \\
  \hline
  {\bf M}^+   & {\bf B_2}
\end{array}
\right)\ .
\end{equation}
\end{mathletters}
The matrix elements and details of the evaluation are
given in the appendix \ref{app:res3}. The main problem
is that one has to find a tractable condition for
\begin{equation}
\label{B2def}
{\bf B_2}^{-1} = {\bf D_2}- {\bf N}^+{\bf D_1}^{-1}{\bf N}
\end{equation}
to be singular. But with expansion tricks similar to the ones
used above this obstacle can be overcome.
For bipartite lattices a relatively simple final formula
is found (\ref{bound4}). An evaluation for the
triangular lattice (appendix \ref{app:tri}) and the
kagome lattice (appendix \ref{app:kagome})  is possible as well.

\section{Results for various lattices} \label{sect:resvarlat}
\subsection{Square lattice} \label{subs:square}

The square lattice represents the simplest bipartite lattice structure
in two space dimensions and has therefore been at the center of interest
in most of the publications dealing with the variational investigation of
Nagaoka
stability\cite{shast90a,basil90,gebha91a,linde91,hanis93,wurth95,wurth96}.
The energy band is given by
\begin{equation} \label{quad_dispers}
\varepsilon_{\Box} (\underline{k}) = -2t \, (\cos k_{x} + \cos k_{y}) \ ,
\end{equation}
with the lattice spacing set to 1. The DOS
$\rho_{\Box} (\varepsilon)$ which is depicted in fig.\ \ref{fig:quad_1}(a)
can be expressed by a complete elliptic integral of the first kind (see
appendix \ref{app:dos}). For positive hopping matrix element $t$ the
lower band edge is reached at $\underline{k}_{\rm b} =
\underline{0}$ while the maxima of the band structure are
located at the corners of the square shaped first Brillouin zone
($\underline{k}_{\rm t}= (\pm \pi, \pm \pi)$). The logarithmic
van Hove singularity at $\varepsilon = 0$ corresponds to the saddle points of
the dispersion (\ref{quad_dispers}). The symmetric shape of the DOS
with respect to $\varepsilon = 0$ reflects the
particle-hole symmetry of the Hubbard model on the square lattice. In the
following we make use of this symmetry and consider only the case of
a less than half filled lattice ($0 \leq n \leq 1$) and $t>0$.
\begin{figure}[htb]
  \setlength{\unitlength}{1cm}
 \begin{picture}(8,13.8)
   \special{psfile=quad_1.ps angle=0 hscale=50 vscale=50
     hoffset=-50 voffset=-12}
 \end{picture} \par
 \caption{ \label{fig:quad_1}
   (a) DOS for the square lattice ($t=1$), (b) spin flip energy at
   $U=\infty$ as a function of the hole density for Gw, NN and RES0
   on the square lattice ($t=1$).}
\end{figure}

Fig.\ \ref{fig:quad_1}(b) shows the spin flip energies at
$U=\infty$ resulting from the variational criteria discussed in the previous
sections  as a function of $\delta$.
The  Gutzwiller wave function ((\ref{ansatz:gw}) with $f=0$)
gives a critical hole density $\delta_{\rm cr} = 0.4905$ for the instability
of the Nagaoka state\cite{shast90a}. For the variational
ansatz (\ref{ansatz:gw}) including nearest neighbor hopping processes
of the majority spins (finite $f$), the spin flip energy is considerably
lowered and the critical hole density decreases to $\delta_{\rm cr} = 0.4155$.
The evaluation the variational state RES0,
which contains \textit{all} spin-up
hopping terms of the Basile-Elser type, leads to $\delta_{\rm cr} = 0.4045$.
Thereby we reproduce up to the fifth digit our result obtained in
\cite{hanis93} where we took into account
hopping processes over a distance of up to four lattice spacings.

The fact that the reduction of the spin flip energy in
fig.\ \ref{fig:quad_1}(b) is mainly due to the nearest neighbor
term demonstrates the overwhelming
importance of \textit{local} polarizations of the spin up Fermi sea for
the instability of the Nagaoka state.
 The resolvent method  treats implicitly an \textit{infinite}
number of variational parameters and
makes it
\begin{figure}[htb]
  \setlength{\unitlength}{1cm}
 \begin{picture}(8,13.8)
   \special{psfile=quad_2.ps angle=0 hscale=50 vscale=50
     hoffset=-50 voffset=-12}
 \end{picture} \par
 \caption{ \label{fig:quad_2}
   (a) Phase diagram ($n<1$): Nagaoka instability lines on the square
   lattice for Gw (dashed-dotted), NN (long dashed), RES1, RES2, RES3
   (full lines, from bottom to top), and the 1100 parameter ansatz of
   Wurth et al.\cite{wurth96} (short dashed), (b) spin flip energy for
   $U_{\rm red} = 0.8$ and $t=1$ as a function of the hole density for Gw
   (dashed-dotted), NN (dashed), RES1, RES2, and RES3 (full lines,
   from top to bottom).}
\end{figure} \noindent
 possible to investigate the full Basile-Elser ansatz for the first time
in the thermodynamic limit. Compared to the iterative
method used in \cite{hanis93} it has the remarkable advantage that the lowest
possible spin flip energy in a given subspace can be calculated
\textit{without} explicit
knowledge of the corresponding state.
As we will see in section E,
it is not generally true that the best value for $\delta_{\rm cr}$
within the Basile-Elser subspace can be obtained by restricting the spin
up hopping processes to a small cluster centered at the position
of the flipped spin.

Fig.\ \ref{fig:quad_2}(a) shows the Nagaoka instability lines in the
phase diagram for the Gutzwiller single spin flip (Gw), the
nearest neighbor ansatz (NN) (\ref{ansatz:gw})
 as well as for the wave
functions RES1, RES2, and RES3 evaluated by means of the
resolvent method. The $\uparrow$-hopping terms appear to be much
less efficient in suppressing the Nagaoka state if the hole density is
small (because most of the sites near the flipped spin are already occupied by
a spin-up electron) and  the on-site repulsion $U$ is finite (because the
terms  all exclude double occupancies at the down spin position). Since the
Gutzwiller projector (with $g > 0$)  represents the only term
contained in RES1 which is relevant for $U<\infty$, the critical on-site
repulsion near half filling is only slightly increased and $U_{\rm cr}$ remains
finite for $\delta= 0$. A remarkable improvement is obtained by allowing
for nearest neighbor exchange processes and thereby taking
into account the antiferromagnetic
tendency  of the nearly half-filled Hubbard model. This is embodied in the
ansatz RES2. For a constant non-zero value of the DOS
at the upper band edge it leads to the asymptotic
behavior $U_{\rm cr, red} (\delta) = 1 - {\cal O}(\delta)$ for $\delta
\rightarrow 0$. This implies the instability of the Nagaoka state for
all finite values of $U$ in this limit. Fig.\ \ref{fig:quad_2}(b) shows
that the optimum spin flip energy for RES2 plotted as a function
of the hole density for a fixed finite value of $U$ approaches a finite
negative value of the order $t^2/U$ at half filling while it vanishes for
all wave functions containing only the Gutzwiller projector.

The asymptotic behavior for $\delta \rightarrow 0$ of the spin flip energy
and of the Nagaoka instability line $U_{\rm cr} (\delta)$ is not affected by the
extension of the Hilbert subspace to the full resolvent ansatz RES3. As
for $U=\infty$ the \textit{local} terms play the most important role in
destabilizing Nagaoka ferromagnetism.
With increasing hole density exchange processes become less important
and the Nagaoka instability lines for RES2 and RES3 approach the one obtained
for RES1. Since \textit{all} RES wave functions differ only in the
subspace with double occupancies the
corresponding instability lines end up with a diverging on-site
repulsion $U_{\rm cr}$ at  the critical hole density
$\delta_{\rm cr} = 0.4045$  obtained for RES0.

Fig.\ \ref{fig:quad_2}(a) displays also the best known variational
bound for the Nagaoka stability regime on the square lattice computed by
Wurth et al.\ \cite{wurth96}. The corresponding state contains 1100 terms,
most of them describing excitations
of the spin-up Fermi sea with up to two particle-hole pairs located within
a 9$\times$9 plaquette around the down spin position. The critical hole
density obtained with this variational wave function is $\delta_{\rm cr} = 0.2514$
and the minimum critical on-site repulsion is $U_{\rm cr}^{\rm min}/t = 77.74$
(RES3: $U_{\rm cr}^{\rm min}/t = 36.21$). Comparing these results one should keep
in mind that the resolvent method allows to derive analytic expressions
for the Nagaoka instability line $U_{\rm cr} (\delta)$, at least for RES1 and
RES2, while the calculation of the phase boundary for the 1100 parameter state
requires an immense numerical effort.

\subsection{Square lattice with next-nearest neighbor hopping
\label{subs:tprime}}

Extending the Hamiltonian (\ref{def:model}) by taking next-nearest
neighbor hopping processes of the electrons into account and introducing
a corresponding hopping amplitude $t'$  allows to \textit{create} a
particle-hole asymmetry of the DOS. Variation of the ratio $t'/t$
makes it possible to simulate a continuous ``transition'' between a bipartite
and a non-bipartite lattice. In this subsection we investigate how this
transition affects the stability of the Nagaoka state with respect to a
Gutzwiller single spin flip on the square lattice. Furthermore we will give
a perturbation argument  for $|t| \ll |t'|$.

The band dispersion of the so-called $t$-$t'$-$U$ model on the square lattice
is given by
\begin{equation} \label{ttstrich_dispers}
\varepsilon_{t-t'} (\underline{k}) = -2t \, (\cos k_x + \cos k_y)
-4t' \cos k_x \cos k_y \ .
\end{equation}
For $t,t'>0$ the lower band edge $\varepsilon_{\rm b}
= -4(t+t')$ is reached at $\underline{k}_{\rm b}
= \underline{0}$. The maxima of the band structure are located at the
corners  of the
Brillouin square for $t' < t/2$ and at the edge centers for $t' > t/2$,
respectively. Exactly for $t'=t/2$ the maximum single particle
energy $\varepsilon_{\rm t} = 2t$ is reached at the whole border
of the Brillouin zone. This leads to a nesting situation
and to the largest possible partice-hole aymmetry with a diverging DOS
at the upper band edge. For $t' > t/2$ local minima of the band structure
develop at the corners of the Brillouin zone leading to a step in the DOS.
In the limit $t/t' \rightarrow 0$ the
single particle energy at these $\underline{k}$-points reaches
the lower band edge. The calculation of the DOS
$\rho_{t-t'} (\varepsilon)$ requires in general a numerical
$\underline{k}$-integration. Only for $t'=t/2$ it is possible to map
$\rho_{t-t'} (\varepsilon)$ on the DOS for $t=0$ and hence on a complete
elliptic integral (see appendix \ref{app:dos}):
\begin{equation} \label{zust_ttstrich}
\rho_{t-t'} (\varepsilon) = \left(1 - \frac{\varepsilon}{2t}\right)^{-1/2}
\ \ \rho_{\Box} \left(2t \sqrt{1 - \frac{\varepsilon}{2t}} \right)
\ .
\end{equation}

The symmetry of the Nagaoka stability regime with respect to half filling
found in the ``pure'' Hubbard model is destroyed if the next-nearest neighbor
hopping $t'$ is switched on. In analogy to the non-bipartite triangular
and kagome lattices (see \cite{hanis95} and sect.\ \ref{subs:tri} in this
paper) one should expect that the tendency towards saturated ferromagnetism
increases for more than half filling and decreases for $n <1$.
The RES ansatzes with the reduction to DOS integrals cannot be used for the
$t$-$t'$ model since the $t'$-hops go beyond nearest neighbor hopping.

The calculation of the optimum spin flip energy for the Gutzwiller ansatz
((\ref{ansatz:gw}) with $f=0$)
 requires additional effort for the $t$-$t'$-$U$ model
due to the more complicated structure of the  band
dispersion (\ref{ttstrich_dispers}). The kinetic energy of the
flipped spin no longer depends only on $\varepsilon_{\rm b}$ but
also on the corresponding \textit{momentum} $\underline{k}_{\rm b}$.
For $t,t' > 0$ (i.e.\ for less than half filling) we find
$\underline{k}_{\rm b} = \underline{0}$ as for $t'=0$, whereas
for  $t,t' < 0$ (i.e.\ for more than half filling) we choose
$\underline{k}_{\rm b} = (\pi, \pi)$ for $t'/t \leq 1/2$ and
$\underline{k}_{\rm b} = (\pi, 0)$ for $t'/t > 1/2$.

Fig.\ \ref{fig:tstrich_1} shows the DOS for the
\begin{figure}[htb]
 \setlength{\unitlength}{1cm}
 \begin{picture}(8,13.8)
   \special{psfile=tstrich_1.ps angle=0
           hscale=50 vscale=50 hoffset=-50 voffset=-12}
 \end{picture} \par
\caption{ \label{fig:tstrich_1} $t$-$t'$-$U$ model on the square lattice for
$|t'| \leq |t|/2:$ $|t'| = 1 - |t| = 0,\, 0.1,\, 0.2,\, 0.3,\,
 1/3$:
(a) DOS $\rho_{t-t'}(\varepsilon)$ for $t, t' > 0$ ($\rho_{t-t'}(\varepsilon)$
    for $t, t' < 0$ is obtained by $\varepsilon \leftrightarrow -\varepsilon$),
(b) Nagaoka instability lines for a Gutzwiller single spin flip (the curves
    for $n<1$ correspond to $t, t' >0$ whereas the curves for $n>1$
    correspond to $t, t'<0$).}
\end{figure} \noindent
$t$-$t'$-$U$ model on the square lattice and the corresponding Nagaoka
instability lines in the phase diagram for various ratios $t'/t \leq 1/2$.
We set $|t| + |t'| = 1$ so that the lower band edge is always at
$\varepsilon_{\rm b} = -4$. Increasing $t'/t$ leads to a
lower DOS at $\varepsilon_{\rm b}$ and a higher DOS
at $\varepsilon_{\rm t}$, while  the logarithmic singularity at
$\varepsilon = 4 t'$ approaches the upper band edge. The maximum particle
hole asymmetry is reached at $|t'/t| = 1/2$ (i.e.\ $|t'|=1/3$) where the DOS
(\ref{zust_ttstrich}) diverges like
$(\sqrt{\varepsilon_{\rm t} - \varepsilon}
\, \cdot \, |\log (\varepsilon_{\rm t} - \varepsilon)|)^{-1}$
for $\varepsilon \approx \varepsilon_{\rm t}$.
The Nagaoka stability region for less than half filling shrinks as
$t'/t$ is increased and disappears at $t'/t = 1/2$ (Figs.\
\ref{fig:tstrich_1}(b), \ref{fig:tstrich_3}). On the other hand, it expands
rapidly for $n>1$, especially in the limit $n=2$. At $t'/t =1/2$  the
Nagaoka state is stable
towards a Gutzwiller single spin flip for all $U>0$ in this limit. Even
the slope of the Nagaoka instability line $U_{\rm cr} (n)$ vanishes at $n=2$.

If one increases the ratio $t'/t$ beyond 1/2, the logarithmic
singularity in the DOS is gradually shifted back towards
$\varepsilon = 0$ and the shape of $\rho_{t-t'} (\varepsilon)$ becomes more
and more symmetric (fig.\ \ref{fig:tstrich_2}(a)).
\begin{figure}[htb]
 \setlength{\unitlength}{1cm}
 \begin{picture}(8,13.8)
   \special{psfile=tstrich_2.ps angle=0
           hscale=50 vscale=50 hoffset=-50 voffset=-12}
 \end{picture} \par
\caption{ \label{fig:tstrich_2} $t$-$t'$-$U$ model on the square lattice for
$|t'| \geq |t|/2:$ $|t'| = 1 - |t| = 1/3,\, 0.4,\, 0.5,\, 2/3,\, 1$:
(a) DOS $\rho_{t-t'}(\varepsilon)$ for $t, t' > 0$ ($\rho_{t-t'}(\varepsilon)$
    for $t, t' < 0$ is obtained by $\varepsilon \leftrightarrow
    -\varepsilon$),
(b) Nagaoka instability lines for a Gutzwiller single spin flip (the curves
    for $n<1$ correspond to $t, t' >0$ whereas the curves for $n>1$
    correspond to $t, t'<0$).}
\end{figure} \noindent
Nevertheless  the step at $\varepsilon = 4t'(1-t/t')$ remains present for all
$t/t' >0$. The DOS at $t=0$ is identical to
$\rho_{\Box} (\varepsilon)$, which reminds us that the $t'$-$U$ model with
suppressed nearest neighbor hopping consists of \textit{two}
completely decoupled square lattices.

At $t'=t$ the Nagaoka stability region in the phase diagram is found to
be still very asymmetric with respect to $n=1$ (fig.\ \ref{fig:tstrich_2}(b)).
A further increase of $t'/t$ makes the phase boundaries above and below
half filling approach the ones obtained at $t=0$.
Within our variational calculations, the local stability of the
saturated ferromagnetic state is identical in both limiting cases
$t' = 0$ and $t =0$, but see the
perturbative argument below. The step in the DOS, however, leads
to a cusp in the Nagaoka stability line $U_{\rm cr} (n)$ for all $|t/t'| <2$.
For $t \rightarrow 0$, this cusp approaches $n=1$ and
$dU_{\rm cr}/dn|_{n=1+}$ is discontinuous at $t=0$.
This represents a qualitative
difference to the limit $t' \rightarrow 0$.

In fig.\ \ref{fig:tstrich_3} the upper and lower critical densities  for the
\begin{figure}[htb]
 \setlength{\unitlength}{1cm}
 \begin{picture}(8,6)
   \special{psfile=tstrich_3.ps angle=0
           hscale=50 vscale=50 hoffset=-50 voffset=-210}
 \end{picture} \par
 \caption{ \label{fig:tstrich_3} Critical densities for the Nagaoka instability
at $U=\infty$ for a Gutzwiller single spin flip on the square lattice as a
function of $|t'| = 1 - |t|$. Between the two full lines the Nagaoka state
is found to be possibly stable. The dashed-dotted line marks the singular case
$|t'/t|=1/2$ where the particle-hole asymmetry reaches its maximum.}
\end{figure} \noindent
Nagaoka instability at $U=\infty$ are plotted as functions of
$|t'| = 1 - |t|$. Thereby we once again demonstrate the shift of the
Nagaoka stability region towards more than half filling with increasing
particle-hole asymmetry in the DOS. The regimes
of complete Nagaoka stability for $n>1$ ($-0.21 \leq t' \leq -0.39$) and
of complete Nagaoka instability for $n<1$ ($1/3 \leq t' \leq 0.45$) are
\textit{not} symmetric with respect to $|t'| = 1/3$. There are two different
reasons for this asymmetry. First, since the increase of the DOS
at the lower band edge is more pronounced for $t' \searrow -1/3$ (that is,
on the left hand side of the dashed-dotted line in fig.\ \ref{fig:tstrich_3})
than for $t' \nearrow -1/3$, also the tendency towards saturated
ferromagnetism in the low density limit (corresponding to $n \rightarrow 2$ in
fig.\ \ref{fig:tstrich_3}) is stronger in the former case. Second, the
Nagaoka instability condition near half filling is essentially determined by
the ratio $\varepsilon_{\rm t}/(zt)$, i.e.\ by the asymmetry of
the band edges with respect to $\varepsilon = 0$. The
fact that the latter asymmetry is more pronounced for $|t'|>1/3$ than for
$|t'| < 1/3$ is responsible for the instability of the Nagaoka state
for less than half filling on the right hand side of the dashed-dotted
line in fig.\ \ref{fig:tstrich_3}.

In the limit $t\to 0$, a perturbative arguments gives further insight
in the stability of saturated ferromagnetism. Starting point is the observation
that at $t=0$ the square lattice decomposes into two independent
 square lattices tilted by 45$^\circ$ with hopping element $t'$.
Without any $t$ the two independent Nagaoka states on
each sub-lattice can be oriented arbitrarily without influencing
the energy. Thus we deal with a degenerate situation and investigate by
$E^{(2)}$  (second order perturbation coefficient in $t$)
whether the parallel or the antiparallel orientation is
favored. The linear order $E^{(1)}$ vanishes for particle-hole symmetry
reasons and does not lift the degeneracy.

For the parallel configuration it is straightforward to calculate $E^{(2)}$.
 Without loss of generality  we choose $t'=1/4$ and consider
$\varepsilon({\underline k}) = \varepsilon_0({\underline k}) -
 2t(\cos(k_x)\pm\cos(k_y))$
with $\varepsilon_0({\underline k}) = -\cos(k_x)\cos(k_y)$
as dispersion. The plus sign refers to $n<1,t'>0$ and the minus sign
to $n>1,t'>0$. This can be seen by means of a particle-hole transformation
and a sign transformation $c_{\underline i} \to -c_{\underline i}$ on all sites
with {\em even} $x$-coordinate.
One obtains at constant filling $E^{(2)} = - |\Lambda|/(2t')
A_\pm(\varepsilon_{\rm F})$ with
\begin{eqnarray}\nonumber
&&A_\pm(\varepsilon_{\rm F}) =\\ \nonumber
&& \int\limits_{-\pi}^\pi \frac{d^2k}{(2\pi)^2}
(\cos(k_x)\pm\cos(k_y))^2 \delta(\varepsilon_{\rm F} + \cos(k_x)\cos(k_y))
\\ \label{hamueha}
& =& -\frac{4}{\pi^2}\left(\pm \varepsilon_{\rm F}
K(1-\varepsilon_{\rm F}^2) - E(1-\varepsilon_{\rm F}^2)
\right)
\end{eqnarray}
yielding the dotted curves in fig.\ \ref{fig:tstrich_4}. The relation
\begin{figure}[htb]
 \setlength{\unitlength}{1cm}
 \begin{picture}(8,6)
   \special{psfile=tstrich_4.ps angle=0
           hscale=50 vscale=50 hoffset=-50 voffset=-210}
 \end{picture} \par
 \caption{ \label{fig:tstrich_4} Second order perturbation coefficient
 $E^{(2)}$ in $t$
in units of $4t'N$. Dotted line: for parallel Nagaoka states
(or global ferromagnetic state, see text);
 Dashed line: upper bound to $E^{(2)}$ for
antiparallel Nagaoka states (or global antiferromagnetic state).}
\end{figure} \noindent
(\ref{hamueha}) is found with the help of the quantities $I_n$ in
appendix A of Hanisch/M\"uller-Hartmann\cite{hanis93}; $K$ and $E$
are complete elliptic integrals. Note that the
coefficient $E^{(2)}$ is not continuous across $n=1$.

Next we assess the energy  of two antiparallel Nagaoka states on each of the
sub-lattices. Let us use $a^+_{{\underline k},\sigma}$
for the fermions on the
A sub-lattice and  $b^+_{{\underline k},\sigma}$ for the fermions on the
B sub-lattice. The perturbation reads then
\begin{equation}
H_1 = -2t\sum\limits_{{\underline k}\in  {\rm MBZ}}(\cos(k_x)\pm\cos(k_y))
(a^+_{{\underline k},\sigma} b^{\phantom +}_{{\underline k},\sigma}
 + b^+_{{\underline k},\sigma} a^{\phantom +}_{{\underline k},\sigma})
\end{equation}
where MBZ is the magnetic Brillouin zone.
The second order energy lowering is
\begin{equation}
\label{perturb1}
E^{(2)} t^2 |\Lambda| = -\left\langle A\uparrow, B\downarrow \left|
H_1 (H_0-E_0)^{-1} H_1 \right| A\uparrow, B\downarrow\right\rangle ) \ .
\end{equation}
The acronyms $A\uparrow$ and $B\downarrow$ stand for the
respective Fermi seas.
There are two processes which contribute equally to (\ref{perturb1}).
Either a fermion is shifted from $A$ to $B$ and back or a fermion is
shifted from $B$ to $A$ and back. The latter yields explicitly
\begin{eqnarray}
&&E^{(2)} = - \frac{8}{|\Lambda|} \nonumber
\sum\limits_{{\underline k}\in  {\rm MBZ}}(\cos(k_x)\pm\cos(k_y))^2
\Theta(\varepsilon_{\rm F}-\varepsilon_0({\underline k})
\\
&& \qquad \left\langle A\uparrow \left|
a^{\phantom +}_{{\underline k},\uparrow}
(H_{0,A}-E_{0,A} - \varepsilon({\underline k}))^{-1}
a^+_{{\underline k},\uparrow}
\right|A\uparrow \right\rangle
\\
&& = 4\int\limits \frac{d^2k}{(2\pi)^2}
(\cos(k_x)\pm\cos(k_y))^2 g_{\underline k}(\varepsilon_0({\underline k})
\Theta(\varepsilon_{\rm F}-\varepsilon_0({\underline k})\nonumber
\end{eqnarray}
where $g_{\underline k}$ is the one-particle Green function. Now we specify
that we work at $U=\infty$ and we {\em assume} that  the Nagaoka state
is stable for $t=0$ at the filling considered. If the Nagaoka state is not
stable we do not need to make the present comparison anyway.
Based on our assumption, the Green function is
purely real and negative. It obeys the inequality
\begin{mathletters}
\begin{eqnarray}\label{abschaetz}
g_{\underline k}(\varepsilon_0({\underline k})) &<&
(\varepsilon_0({\underline k})- e_{\underline k})^{-1} <0
\\ \nonumber
e_{\underline k} &:=& \langle a^{\phantom +}_{{\underline k},\uparrow}|
H_{0,A}-E_{0,A}
|a^+_{{\underline k},\uparrow} \rangle
\\ \label{approx_ska}
&=& -e_1/\delta + \varepsilon_0({\underline k}) \delta(1-(e_1/\delta)^2) \ .
\end{eqnarray}
\end{mathletters}
The  estimate (\ref{abschaetz}) corresponds to a simple Gutzwiller ansatz
\cite{shast90a} and yields (\ref{approx_ska}) (see (4) and (5) with
$t=t'=1/4$ and $z=4$ in \cite{hanis93}). Thus we obtain
\begin{equation}
\label{finresper}
E^{(2)} < \frac{4\delta}{f}\int\limits_{-1}^{\varepsilon_{\rm F}}
d\varepsilon \frac{A_\pm(\varepsilon)}{\lambda-\varepsilon}
\end{equation}
where $f= \delta(\delta-1) -e_1^2$, $\lambda=e_1/f$, and
$A_\pm$ from (\ref{hamueha}).
The evaluation of the right hand side of (\ref{finresper})
yields the dashed curves in fig.\ \ref{fig:tstrich_4}.
The essence of fig.\ \ref{fig:tstrich_4}
is that the saturated ferromagnetic state is  unstable in the limit
$t\to 0$ for all fillings. The small region where $E^{(2)}_{\rm FM}$
lies below the upper bound for $E^{(2)}_{\rm AFM}$ does not count
since we know that at these dopings (and for larger dopings)
already  the pure square lattice  at
$t=0$ has no saturated ferromagnetic ground state, see e.g.\
\cite{hanis93,wurth96}.

We wish to draw
the reader's attention to the fact that the comparison in
fig.\ \ref{fig:tstrich_4} is quite different from the main theme of this paper
which is based on single spin flip energies.
Here the global stability is tested with  a completely
different, antiferromagnetic state.   We learn from the perturbative argument
that in fig.\ \ref{fig:tstrich_3} the true lines
$n(t')$ comprising the {\em global} Nagaoka stability region have to converge
both to the point $t'=1,n=1$.

\subsection{Simple cubic lattice}

The energy dispersion of the simple cubic lattice is
\begin{equation} \label{sc_dispers}
\varepsilon_{sc} (\underline{k})
= -2t \, (\cos k_{x} + \cos k_{y} + \cos k_{z}) \ .
\end{equation}
The calculation of $\rho_{sc} (\varepsilon)$ can be performed by
an integration over the known DOS of the square lattice (see appendix
\ref{app:dos}).
The maxima and minima of the energy dispersion (\ref{sc_dispers}) are
$\varepsilon_{\rm t} = z |t|$ and $\varepsilon_{\rm b}
= -z |t|$, respectively, with the coordination number $z = 6$.
At the band edges the DOS (fig.\ \ref{fig:sc_1}) shows the square root
\begin{figure}[htb]
 \setlength{\unitlength}{1cm}
 \begin{picture}(8,6)
   \special{psfile=sc_1.ps angle=0
           hscale=50 vscale=50 hoffset=-50 voffset=-210}
 \end{picture} \par
 \caption{ \label{fig:sc_1} DOS for the simple cubic lattice
 ($t=1$).}
\end{figure} \noindent
behavior which is characteristic for $d=3$.
The van Hove singularities at $\varepsilon = \pm 2t$ correspond to the saddle
points of the dispersion (\ref{sc_dispers}).

Fig.\ \ref{fig:sc_2}(a)
shows the spin flip energy at $U=\infty$ for Gw, NN, and RES0. For small hole
doping the loss of spin-up kinetic energy due to the spin flip is
sufficiently strong to keep the Nagaoka state stable. With increasing
$\delta$ the spin flip energy decreases due to the
gain of kinetic energy for the flipped spin which grows linear with $\delta$
in leading order. The upper bound for the critical
hole density is reduced from $\delta_{\rm cr} = 0.323$ for
Gw\cite{shast90a,mulle91} to $\delta_{\rm cr} = 0.247$ for NN
and finally to $\delta_{\rm cr} = 0.237$ for RES0. As in $d=2$, the NN
hopping
term gives the dominant contribution to the decrease of $\delta_{\rm cr}$
while the extension of the spin-up hopping processes to the whole lattice has
only a small effect.
\begin{figure}[htb]
 \setlength{\unitlength}{1cm}
 \begin{picture}(8,13.8)
   \special{psfile=sc_2.ps angle=0
           hscale=50 vscale=50 hoffset=-50 voffset=-12}
 \end{picture} \par
 \caption{ \label{fig:sc_2} (a) Spin flip energy at $U=\infty$ as a function of
 the hole density on the \textit{sc} lattice ($t=1$) for Gw, NN, and RES0,
(b) phase diagram ($n<1$): Nagaoka instability lines on the
    \textit{sc} lattice  for Gw (dashed-dotted), NN (long
    dashed), RES1 (lower full line), RES2 (short dashed), and RES3
    (upper full line).}
\end{figure}

Roth \cite{roth69} investigated the Nagaoka instability with respect
to a single spin flip on the \textit{sc} lattice already in 1969, making use
of the so-called two pole approximation instead of the projection method.
It was shown later \cite{tan74,allan82} that the Hilbert subspace considered
in \cite{roth69} is equivalent to the Basile-Elser subspace in the limit
$U \rightarrow \infty$. Roth obtained numerically a critical hole
density of $0.24$ which is consistent with our variational result for RES0.

The phase diagram (fig.\ \ref{fig:sc_2}(b)) for the simple cubic lattice
shows a qualitative difference to the square lattice: The critical $U$ at half
filling obtained for the Gutzwiller single spin flip is not at all improved
by including NN hopping terms. Even for RES1
$U_{\rm cr}(\delta=0)$ is still given
by the band width $12 |t|$. This is due to the fact
that for the \textit{sc} lattice the DOS at the upper band edge
vanishes while it is nonzero for the square lattice.

As in $d=2$, the ansatz RES2 leads to $U_{\rm cr}
 (\delta = 0) = \infty$ and to a
considerable reduction of the Nagaoka stability regime near half filling.
For the full resolvent ansatz RES3 we finally achieve a minimum critical
coupling of $U_{\rm cr}^{\rm min} = 48.9 |t|$ (corresponding to $U_{\rm red} = 0.753$)
below which the Nagaoka state is proven to be unstable for all $\delta$.
\begin{figure}[htb]
 \setlength{\unitlength}{1cm}
 \begin{picture}(8,13.8)
   \special{psfile=bcc.ps angle=0
           hscale=50 vscale=50 hoffset=-50 voffset=-12}
 \end{picture} \par
 \caption{ \label{fig:bcc} (a) DOS for the \textit{bcc} lattice ($t=1$).
(b) phase diagram ($n<1$): Nagaoka instability lines on the
    \textit{bcc} lattice  for RES1 (dotted line), RES2 (dashed), and RES3
    (full line).}
\end{figure}\noindent
The region left for a possible Nagaoka ground state on the \textit{sc} lattice
is therefore substantially smaller than on the square lattice (RES3 for the
square lattice: $\delta_{\rm cr}= 0.405, U_{\rm cr}^{\rm min} = 36.2|t|$).
Generally the tendency of the Hubbard model towards a saturated ferromagnetic
ground state on a $d$-dimensional hypercubic lattice becomes weaker with
increasing $d$. M\"uller-Hartmann \cite{mulle93b} showed that the critical
hole density at $U = \infty$ with respect to a Gutzwiller single spin flip
decreases asymptotically as $\delta_{\rm cr} \propto 1/\sqrt{d \, \ln d}$
for $d \gg 1$. In the limiting case of infinite dimensions the ground
state of the Hubbard model is never fully polarized \cite{fazek90}.

\subsection{bcc lattice}
The \textit{bcc} lattice is another interesting example of a
three-dimensional bipartite lattice. It has a slightly higher
coordination number $z=8$ compared to the simple cubic lattice.
Its dispersion reads
\begin{equation}
\varepsilon_{bcc} (\underline{k})
= -8t  \cos( k_{x})\cos( k_{y})  \cos( k_{z}) \ .
\label{bccdispers}
\end{equation}
The calculation of the DOS $\rho_{bcc}(\varepsilon)$ can again be
performed by an integration over the known DOS of the square lattice
(see appendix \ref{app:dos}). The bipartiteness is obvious since
$\varepsilon_{bcc} (\underline{k}+\underline{Q}) +\varepsilon_{bcc}
(\underline{k}) =0$ with $\underline{Q}=(\pi,\pi,\pi)$.
For this reason we consider only $n \le 1$.

The DOS is shown in fig.\ \ref{fig:bcc}(a). The square root
singularities  at the band edges are generic for three dimensions.
The least common  feature for a three dimensional lattice is the
squared logarithmic  singularity at zero energy
$\rho_{bcc}(\varepsilon)\approx \ln^2(\varepsilon)/(4\pi^3)$
which results from the points  in momentum space where all cosines in
(\ref{bccdispers}) vanish, e.g.
$\varepsilon_{bcc} (\underline{k})
\approx -8t (k_x-\pi/2)(k_y-\pi/2)(k_z-\pi/2)$.

Evaluating (\ref{skaband}) for the \textit{bcc} lattice, we find the
critical density  $\delta_{\rm cr}=0.324$ in the Gutzwiller approach.
This is almost the same result as for the simple cubic lattice.
The result $\delta_{\rm cr}=0.239$
for the full ansatz RES0 is also only a tiny bit higher than the RES0
critical doping for the \textit{sc} lattice.  It appears
that the essential ingredients are indeed the dimensionality and
the bipartiteness as we will see below.

The results for finite interaction are shown in fig.\ \ref{fig:bcc}(b).
The value of $U_{\rm BR}$ is $16.413$.  The reduced interaction values
 are very similar to the ones for the simple cubic lattice.
The ansatz RES1 does not capture the diverging interaction for $n\to 1$
but RES2 yields already the asymptotic behavior of RES3 for $n\to 1$.
The critical interaction is $U_{\rm red, cr}= 0.7438$ for RES3.

As far as the local stability of the Nagaoka state is concerned we
do not find any indication that the \textit{bcc} lattice is more
favorable than the simple cubic lattice. Herrmann and Nolting
\cite{herrm97a,herrm97b} found in the framework of the
spectral density approach an enhanced tendency towards ferromagnetism
for the \textit{bcc} lattice.
They investigated the divergence of the susceptibility in the
paramagnetic phase which is enhanced by the large DOS at zero energy.
Combining their result with ours one might come to the
conclusion that the \textit{bcc} lattice favors a non saturated ferromagnetism
for intermediate coupling and doping.

\subsection{Honeycomb lattice}

Besides the square lattice the honeycomb lattice  (see fig.\ 9 in
\cite{hanis95}) is another prominent example of a bipartite lattice in $d=2$.
In contrast to the square lattice it is not a Bravais
lattice, however, but a triangular lattice with a two site basis. The
coordination number is $z=3$ and the band dispersion reads
\begin{equation} \label{honeycomb_dispers}
\varepsilon_{\rm hon} (\underline{k}) = \pm \sqrt{t (3t -
\varepsilon_{\triangle} (\underline{k}))} \ ,
\end{equation}
where $\varepsilon_{\triangle}(\underline{k})$ stands for the energy
dispersion  of the triangular lattice to be described in
(\ref{dreieck_dispers}). Despite this additional complication the
formulae developed in sect.\ \ref{sect:resapp} via the resolvent method
hold here as well (see appendix \ref{app:unfrustrated}).

\begin{figure}[htb]
 \setlength{\unitlength}{1cm}
 \begin{picture}(8,6)
   \special{psfile=honeycomb_1.ps angle=0
           hscale=50 vscale=50 hoffset=-50 voffset=-210}
 \end{picture} \par
 \caption{ \label{fig:honeycomb_1} Spin flip energy at $U=\infty$ as a function
of  the hole density on the honeycomb lattice ($t=1$) for Gw, NN, and RES0.}
\end{figure} \noindent
The instability of the Nagaoka state with respect to Gw and NN
was already discussed in \cite{hanis95}. Here we
present the improvements obtained by the resolvent method.
The evaluation of RES0 shows that hopping processes
with a larger distance from the down spin position have only a very small
influence on the optimum spin flip energy at
$U=\infty$ (fig.\ \ref{fig:honeycomb_1}).

\begin{figure}[htb]
 \setlength{\unitlength}{1cm}
 \begin{picture}(8,6)
   \special{psfile=honeycomb_2.ps angle=0
           hscale=50 vscale=50 hoffset=-50 voffset=-210}
 \end{picture} \par
 \caption{ \label{fig:honeycomb_2} Phase diagram ($n<1$):
Nagaoka instability lines on the honeycomb lattice for Gw (dashed-dotted),
NN (long dashed, almost identical with
RES1), RES2 (short dashed), and RES3 (full line).}
\end{figure}
The
instability gap ($0.379 \leq \delta \leq 0.481$) between the two
possible Nagaoka stability regions remains
almost unchanged compared to the result for NN. The upper
critical hole density is only slightly improved to $\delta_{\rm cr} = 0.643$
from $0.662$ (NN) and $0.802$ (Gw)\cite{hanis95}.

As explained in \cite{hanis95} the Nagaoka stability island in the phase
diagram around quarter filling (fig.\ \ref{fig:honeycomb_2})
 is mainly due
to the zero in the DOS at $\varepsilon = 0$, i.e.\ between the two energy
bands. Since the lattice structure enters the calculation of the optimum
spin flip energy by means of the resolvent method only via the DOS
the stability island is present even for the full resolvent ansatz RES3.
On the other hand, the critical $U$ at half filling diverges for RES2 and
RES3 and the Nagaoka stability region for small $\delta$   shrinks
compared to the results for NN and RES1. These results and the pronounced
difference between the two minimum values of $U$ ($41.2\, |t|$ for the low
doping regime and $17.25\, |t|$ for the stability island) corroborate the
previous conjecture\cite{hanis95} that a saturated ferromagnetic ground state
exists around quarter filling. The lack of a Nagaoka theorem for the honeycomb
lattice\cite{tasak89,hanis95} indicates a degeneracy between the Nagaoka
state and other possible states near half filling  even at $U=\infty$.

\subsection{Triangular lattice} \label{subs:tri}

The triangular lattice is non-bipartite. It can be decomposed into
\textit{three} sub-lattices, each of them having triangular structure.
Investigating the local instability of the Nagaoka state towards a
Gutzwiller single spin flip a Nagaoka ground state was
excluded on the triangular lattice for less than
half filling\cite{mulle91,hanis95}. This is in agreement with the Nagaoka
theorem, which predicts a saturated ferromagnetic ground state
at $U=\infty$ only for the half filled lattice \textit{plus} an additional
electron. Thus we consider henceforth the electron doped case for $t=1$ or,
equivalently, $t=-1$ and $n<1$.

Each lattice site has $z=6$ nearest neighbors located at the corners of a
hexagon. The band dispersion is given by
\begin{equation} \label{dreieck_dispers}
\varepsilon_{\triangle} (\underline{k}) =
2\cos k_x + 4\cos \left(\frac{k_x}{2} \right) \cos \left(
\frac{\sqrt{3} \, k_y}{2} \right)
\end{equation}
where $\underline{k}$ belongs to the likewise hexagon-shaped first
Brillouin zone. The upper band edge ($\varepsilon_{\rm t} = 6$)
is found at the center of the Brillouin zone, whereas the lower band edge
$\varepsilon_{\rm b} = -3$ is reached at the corners of the
hexagon. The DOS (see appendix \ref{app:dos} and fig.\ 1 in \cite{hanis95}),
which can be expressed by a complete elliptic integral, displays a
logarithmic van Hove singularity at $\varepsilon = -2$.
For $\varepsilon_{\rm F}=-2$
 the Fermi surface forms a hexagon with an area of 3/4 of the
whole  Brillouin zone. As usual in $d=2$ the DOS at the band edges is
nonzero ($\rho_b = 4 \rho_t = (\sqrt{3}\pi)^{-1}$).

In contrast to the square lattice, the Nagaoka state remains stable
towards Gw for all fillings $n>1$
at $U=\infty$\cite{shast90a,mulle91}. The corresponding spin flip energy as a
function of $\delta$ is depicted in fig.\ \ref{fig:dreieck_1}(a). Evaluating
NN, however, a negative spin flip energy is found above
$\delta_{\rm cr} = 0.912$ proving the instability of the Nagaoka state
in the low density limit. The resolvent ansatz RES0 lowers the spin flip energy
further and implies $\delta_{\rm cr} = 0.824$ (fig.\ \ref{fig:dreieck_1}(a)).
The difference $\Delta \delta = 0.088$ between the results
obtained for NN and RES0 is eight times larger than the one for  the square
lattice ($\Delta \delta = 0.011$).
\begin{figure}[htb]
 \setlength{\unitlength}{1cm}
 \begin{picture}(8,13.8)
   \special{psfile=dreieck_1.ps angle=0
           hscale=50 vscale=50 hoffset=-50 voffset=-12}
 \end{picture} \par
 \caption{ \label{fig:dreieck_1} (a) Spin flip energy at $U=\infty$ as
a function of the hole density on the triangular lattice ($t=-1$)
for Gw, NN, and RES0,
(b) spin flip energy at $U_{\rm red} = 0.4$ as a function of
the hole density on
the triangular lattice ($t=-1$) for RES1 (dashed-dotted line), RES2
(upper full line), and RES3 (lower full line).}
\end{figure} \noindent
 This  demonstrates the importance
of the spin-up hopping  processes for the instability of Nagaoka ferromagnetism
on the triangular lattice. The reason is that due to the large hole densities
under consideration, the probability to find unoccupied sites near the
flipped spin is quite high. The same line of reasoning applies also for
$U<\infty$, see fig.\ \ref{fig:dreieck_2}.
\begin{figure}[htb]
 \setlength{\unitlength}{1cm}
 \begin{picture}(8,6)
   \special{psfile=dreieck_2.ps angle=0
           hscale=50 vscale=50 hoffset=-50 voffset=-210}
 \end{picture} \par
 \caption{ \label{fig:dreieck_2} Phase diagram ($n>1$):
Nagaoka instability lines on the triangular lattice for Gw (dashed-dotted),
NN (lower full line), RES1
(short dashed), RES2 (long dashed), and RES3 (upper full line).}
\end{figure} \noindent

Previously we investigated a variational state which restricts
the hopping processes to a 31-site cluster around the position of the
flipped spin\cite{hanis95} and obtained $\delta_{\rm cr} = 0.887$.
Although the nearest neighbor processes once again are the most important
ones, the number of relevant hopping processes
on the triangular lattice turns out to be much larger than on the square
lattice. Hence the critical hole density $\delta_{\rm cr} = 0.824$
found by the resolvent method is
essentially lower than the one found from the finite cluster calculations.
 Moreover, the evaluation of
RES0 requires much less analytical and numerical effort than the iterative
extension of the variational ansatz by additional hopping processes.
For details on the application of the resolvent method to the triangular
lattice see appendix \ref{app:tri}.

Near half filling the influence of the majority spin hopping processes
contained in NN and RES1 (which suppress double occupancies) on the Nagaoka
stability is negligible, as expected (fig.\ \ref{fig:dreieck_2}).
In contrast to this the resolvent ansatz RES2 with nearest neighbor
hopping processes \textit{creating} double occupancies
leads to a negative spin flip energy near half filling for all
$U<\infty$ and hence to a divergence of $U_{\rm cr} (n=1)$
(fig.\ \ref{fig:dreieck_2}). It turns out,
however, that for larger hole densities, when the exchange effect looses its
importance,  RES2 is somewhat \textit{less} successful than RES1.
The plot of the spin flip energy as  a function of $\delta$ for the
comparatively small on-site repulsion $U_{\rm red} = 0.4$ in fig.\
\ref{fig:dreieck_1}(b) demonstrates that above $\delta \simeq 0.12$
the creation of extra holes near the flipped spin as described by RES2 is
energetically unfavorable. The full resolvent ansatz RES3, comprising RES1
and RES2, gives of course the best lower bound for the Nagaoka instability
line $U_{\rm cr} (\delta)$. The minimum critical coupling obtained for RES3
is $U_{\rm cr}^{\rm min} = 9.62 |t|$ ($U_{\rm red} = 0.378$),
 the critical hole density
at $U=\infty$ is given by the RES0 value $\delta_{\rm cr} = 0.824$. Hence the
region for a possible Nagaoka ground state on the triangular lattice
appears to be much larger than on the bipartite square and honeycomb
lattices.

\subsection{Kagome lattice}

Taking the kagome lattice as an example of a frustrated non-Bravais lattice
we want to demonstrate that the resolvent method works also for this class
of lattices. Representing the line graph \cite{anmerklinegraph} of the
honeycomb lattice the kagome lattice (for $t<0$) shows a flat, i.\ e.\
dispersionless band with spectral weight 1/3 at the lower band edge
$\varepsilon_{\rm b} = -2 |t|$ (fig.\ \ref{fig:kagome_1}).
\begin{figure}[htb]
 \setlength{\unitlength}{1cm}
 \begin{picture}(8,6)
   \special{psfile=kagome_1.ps angle=0
           hscale=50 vscale=50 hoffset=-50 voffset=-210}
 \end{picture} \par
 \caption{ \label{fig:kagome_1} DOS for the kagome lattice
($t=-1$).}
\end{figure} \noindent
 All line graphs display such a flat band\cite{mielk91a}.
 The kagome lattice is
the first and the most prominent example of so-called flat-band
ferromagnetism\cite{mielk92,tasak95}.
A macroscopic degeneracy of the lowest single
particle energy leads for certain band fillings to a unique saturated
ferromagnetic ground state. Mielke \cite{mielk91a}
 proved that the Nagaoka state
is the unique ground state of the Hubbard model on the kagome lattice
for all $U>0$ at $n = 1/3$.
Although in the flat-band regime every ground state of the
Hamiltonian (\ref{def:model}) is a simultaneous eigenstate of $H_{\rm kin}$ and
$H_{\rm pot}$, the uniqueness of the ground state is not trivial.
For $n <  1/3$ the fully polarized ground state is not unique
\cite{mielk93b}.

The kagome lattice can be considered as a triangular lattice with a basis
of three lattice points\cite{hanis95}, see also appendix
\ref{app:kagome}. Besides the flat band
$\varepsilon (\underline{k}) = 2t$ the diagonalization of
$H_{\rm kin}$ leads to the two dispersive bands
\begin{equation}
\label{kagdispers}
\varepsilon_{K} (\underline{k}) = -t \left( 1 \pm \sqrt{3 -
\varepsilon_{\triangle} (\underline{k})/t} \right)
\end{equation}
where $\varepsilon_{\triangle} (\underline{k})$ stands for the dispersion
(\ref{dreieck_dispers}) of the triangular lattice.
For the kagome lattice the resolvent method requires
less effort than for the triangular lattice with $t<0$ since the lower
band edge $\varepsilon_{\rm b} = -2 |t|$ is reached at
$\underline{k}_{\rm b} =  \underline{0}$ for one
dispersive band and for the flat band of course. Thus  $\underline{q} =
 \underline{0}$ is the optimum momentum as for bipartite lattices.
Hence all
lattice dependent quantities appearing in our formulae can be calculated
as integrals over the  DOS $\rho_{K} (\varepsilon)$.
 For $t=-1$ one finds the DOS of the kagome
lattice (see fig.\ \ref{fig:kagome_1} and appendix \ref{app:dos}) as
\begin{equation}
\rho_{K} (\varepsilon) = \frac{1}{3} \, \delta(\varepsilon +2) +
\frac{2}{3} | \varepsilon - 1| \cdot \rho_{\triangle} \left(
(\varepsilon -1)^2-3 \right);
\end{equation}
$\rho_{\triangle} (\varepsilon)$ is the DOS
of the triangular lattice. Nevertheless, the fact that
the kagome lattice is \textit{not} a Bravais lattice induces
some changes in the analytic expressions for the spin flip energy
(appendix \ref{app:kagome}).

\begin{figure}[htb]
 \setlength{\unitlength}{1cm}
 \begin{picture}(8,13.8)
   \special{psfile=kagome_2.ps angle=0
           hscale=50 vscale=50 hoffset=-50 voffset=-12}
 \end{picture} \par
 \caption{ \label{fig:kagome_2} Spin flip energy at $U=\infty$ as a function
of the hole density on the kagome lattice for Gw (dashed-dotted), NN (dashed),
and RES0 (a) for $t=1$,  (b) for $t=-1$.}
\end{figure}
Fig.\ \ref{fig:kagome_2}(b)
 shows the spin flip energy for $U=\infty$ and
$t=-1$ as a function of $\delta$  for RES0 compared to the Gw and NN
results obtained in \cite{hanis95}. As for the honeycomb lattice the
enhancement of the Nagaoka stability for $\delta \rightarrow 1/3$ is due to
the zero in the DOS. The effect of the additional spin-up hopping processes
contained in RES0 is most pronounced for $\delta > 1/3$. But the spin flip
energy remains positive for all band fillings. Note that the exact result
in the flat-band regime is a zero spin flip energy\cite{mielk93b,hanis95}.
The phase diagram for $n>1$ in fig.\ \ref{fig:kagome_3}
\begin{figure}[htb]
 \setlength{\unitlength}{1cm}
 \begin{picture}(8,6)
   \special{psfile=kagome_3neu.ps angle=0
           hscale=50 vscale=50 hoffset=-50 voffset=-210}
 \end{picture} \par
 \caption{ \label{fig:kagome_3} Phase diagram: Nagaoka instability lines on the
kagome lattice for Gw, NN, RES1 (lower full
line), RES2, and RES3 (upper full line). For $n<1$ the
difference between RES1, RES2, and RES3 is less than the line width.}
\end{figure} \noindent
 shows a
strong tendency towards Nagaoka ferromagnetism also beyond the flat-band
regime, where we find the Nagaoka state to be stable for all $U>0$.
There is only a marginal difference between the Nagaoka instability lines
for NN and for RES1, since the values of $U$  under consideration are too
small to allow a significant reduction of the spin flip energy
by Basile-Elser hopping processes. Near half filling, however, we are able to
restrict the Nagaoka stability region by RES2, i.e.\ by taking
antiferromagnetic exchange processes into account.
As for the triangular lattice, for a certain range of filling around
$n = 3/2$ away from half filling RES2 gives a weaker bound for
$U_{\rm cr} (\delta)$ than RES1.

For positive hopping matrix element $t$ the flat band is found at the
\textit{upper} band edge. The flat-band regime for $n<1$  corresponds to
hole densities $0 \leq \delta \leq 1/3$. Since
$\underline{k}_{\rm b} = \underline{0}$ \textit{and}
$\varepsilon_{\rm b} = -zt$ the resolvent method formulae
are those of the bipartite lattices (see appendix \ref{app:unfrustrated}).
Figs.\ \ref{fig:kagome_2}(a) and \ref{fig:kagome_3} show that the small
Nagaoka stability island found previously\cite{hanis95} for very large $U$
around quarter filling is still present for RES0 -- RES3. The upper
critical hole density is reduced from 0.727 (NN) to 0.715 (RES0) and
$U_{\rm cr}^{\rm min}$ reaches $191.5 |t|$ (RES3)
instead of $129.5 |t|$ (NN) though.
These results may indicate that this stability island really provides an
example of a saturated ferromagnetic ground state on a non-bipartite lattice
for less than half filling. Its origin\cite{hanis95} is the zero in the DOS
enhancing the stability of the Nagaoka state around $\delta = 2/3$.

\subsection{fcc and hcp lattices}
The \textit{fcc} and \textit{hcp} lattices as the most prominent
close-packed lattices in $d=3$  are found in numerous real substances among
them the ferromagnetic transition metals Ni (\textit{fcc}) and
Co (\textit{hcp}). The face-centered cubic lattice is
a Bravais lattice with coordination number $z=12$.  Its band dispersion
\begin{eqnarray} \nonumber
&&\varepsilon_{fcc} (\underline{k}) =\\
\label{fcc_dispers}
&& \quad -4t \, (\cos k_x \cos k_y + \cos k_x
\cos k_z + \cos k_y \cos k_z)
\end{eqnarray}
is related to the dispersion (\ref{sc_dispers}) of the simple cubic
(\textit{sc}) lattice via\cite{mulle91}
\begin{equation} \label{ident:sc_fcc}
\varepsilon_{fcc} (\underline{k}) =
- \frac{\varepsilon_{sc}^2 (\underline{k})}{2t}
-\frac{\varepsilon_{sc} (2\underline{k})}{2} + 3t \ .
\end{equation}
The hexagonal close-packed lattice (also with $z = 12$) is not a Bravais
lattice but a hexagonal lattice with basis.
\begin{figure}[htb]
 \setlength{\unitlength}{1cm}
 \begin{picture}(8,13.8)
   \special{psfile=fcc.ps angle=0
           hscale=50 vscale=50 hoffset=-50 voffset=-12}
 \end{picture} \par
 \caption{ \label{fig:fcc} (a) Identical DOS for the \textit{fcc} and
\textit{hcp} lattices ($t=-1$) and DOS for the \textit{fcc} lattice with
additional next nearest neighbor hopping ($t' = t/2$),
(b) phase diagram ($n>1$): Nagaoka instability lines on the \textit{fcc}
lattice for Gw and NN and on the $t$-$t'$-\textit{fcc} lattice with $t'=t/2$
for Gw.}
\end{figure} \noindent
 Within the hexagonal planes,
which we assume to be parallel to the $xy$ plane,
the $\underline{k}$-dependence of the two energy bands reduces
to the energy dispersion (\ref{dreieck_dispers}) of the triangular
lattice\cite{mulle91}:
\begin{eqnarray} \nonumber
&&\varepsilon_{hcp} (\underline{k}) = \varepsilon_{\triangle} (k_x, k_y) \\
&& \label{dispers_hcp} \qquad \pm
2t \cos \left(\sqrt{2/3}\, k_z \right) \cdot \sqrt{3 - \varepsilon_{\triangle}
(k_x, k_y)/t} \ .
\end{eqnarray}
Hence it is possible to compute the DOS of the \textit{hcp} lattice by
integration over $\rho_{\triangle} (\varepsilon)$ (see appendix \ref{app:dos}).

Modelling the \textit{fcc} and \textit{hcp} structures with close-packed
spheres, the sequence of layers with different positions of the sphere
centers is known to be ABABAB... for the \textit{hcp}
and ABCABC... for the \textit{fcc} lattice. Diagonalizing the kinetic part
of the  Hamiltonian (\ref{def:model}) in each of the hexagonal
planes it turns out that
the terms reflecting the different arrangement of the planes disappear if
one chooses the Fourier transformation in a convenient way. The
\textit{densities of states} for the \textit{fcc} and for the \textit{hcp}
lattices are  therefore \textit{identical}
as well as our variational results on the stability of the Nagaoka state
with respect to Gw and NN.

For less than half filling a saturated ferromagnetic ground state was
already excluded due to the complete instability of the Nagaoka state towards
Gw at $U=\infty$\cite{mulle91}. Therefore we only
investigate the case of more than half filling which corresponds to $t<0$.
The DOS (fig.\ \ref{fig:fcc}(a))
displays  the $d=3$ square root behavior
$\rho (\varepsilon) \propto \sqrt{\varepsilon_{\rm t} - \varepsilon}$
at the upper band edge $\varepsilon_{\rm t} = z |t|$. For the \textit{fcc}
lattice the lower
band edge $\varepsilon_{\rm b} = -4|t|$  is reached on different
\textit{lines} in $\underline{k}$-space which intersect in several critical
points located on the border of the Brillouin zone. This reduces the
``effective dimensionality'' of the van Hove singularity by one
and leads to a \textit{logarithmic singularity} in the DOS.
As a consequence, the Nagaoka state remains stable in the low density limit
(corresponding to $n \rightarrow 2$ for $t>0$) for all $U>0$ with respect to
Gw and NN (fig.\  \ref{fig:fcc}(b)). This result indicates the strong
tendency towards
Nagaoka ferromagnetism on the \textit{fcc} lattice, especially in comparison
with the triangular lattice where we proved the \textit{instability} of the
Nagaoka state even for $U=\infty$ in the low density  limit.
Also for low and intermediate hole doping the extension of the Gutzwiller
wave function by nearest neighbor hopping processes yields only a slight
reduction of the Nagaoka stability region in the phase diagram.  This is in
sharp contrast, for example, to the situation on the \textit{sc} lattice.
The resolvent method was not applied to the \textit{fcc} and \textit{hcp}
lattices since three dimensional momentum integrals would have to be performed
in order to calculate $h(\omega)$ and $\overline{h} (\omega)$ (see
appendix \ref{app:tri}).

As for the square lattice (see section \ref{subs:tprime}) the
particle-hole asymmetry of the
DOS is even enhanced if one extends $H_{\rm kin}$ by electron
hopping between \textit{next nearest neighbor sites}
with a hopping amplitude $t'$.
On the \textit{fcc} lattice, these sites form a simple cubic
structure such that the additional contribution to the dispersion exactly
compensates the second term on the right hand side of (\ref{ident:sc_fcc})
if $t' = t/2$. In this case the DOS for the $t$-$t'$-\textit{fcc} lattice
is connected to the DOS of the \textit{sc} lattice  via
\begin{equation} \label{ident:ttstrich}
\rho_{t-t'} (\varepsilon) = \sqrt{\frac{2}{3(1-\frac{\varepsilon}{3t})}} \
\cdot \ \rho_{sc} \left(\sqrt{6(1 - \frac{\varepsilon}{3t})t} \right)
\end{equation}
and therefore finally simplifies to an integral over
$\rho_{\Box} (\varepsilon)$ (see appendix \ref{app:dos}). The next nearest
neighbor hopping with amplitude $t' = t/2$ creates a \textit{square root}
divergence of the DOS at the lower band edge (for $t,t' <0$)
in contrast to the \textit{logarithmic} singularity obtained for $t'=0$
(see fig.\ \ref{fig:fcc}(a)).

The Nagaoka instability line for a Gutzwiller single spin flip
on the \textit{fcc} lattice with $t' = t/2$ is compared
with the result for the simple \textit{fcc} lattice ($t'=0$)
in fig.\ \ref{fig:fcc}(b). The more pronounced singularity of the DOS at the
lower band edge leads to an even more pronounced stability of the Nagaoka
state in the low density limit. We find $U_{\rm cr} (\delta)
\propto 1 - \delta$
instead of $U_{\rm cr} (\delta) \propto 1/\log(1-\delta)$ for $t' =0$.
The slight increase of the critical $U$ at half filling is due to the
different band width of the $t$-$t'$-$U$ model ($18|t|$ instead of $16|t|$).

\section{Conclusions} \label{sect:conclus}

In summary, we investigated the stability of the Nagaoka state for
a series of  two- and three-dimensional lattices:
 the square $t$-, the square $t$-$t'$-, the simple cubic, the \textit{bcc},
the honeycomb,
the triangular, the kagome, and the \textit{fcc} (\textit{hcp}) lattice.
The results were mostly variational in nature and concerned the energy
change due to a single spin flip. By the resolvent approach
the eigenvalue problem in the
variational subspace was reduced  to a matrix
inversion problem. The relatively simple structure of the matrices
under consideration permits to convert the matrix inversion into a
scalar inversion (or the inversion of a $2\times2$ matrix).
For the $t$-$t'$ square lattice a perturbative approach in $t$ was
used as well for investigating the $t/t'\to 0$ limit.

The ansatzes RES0 - RES3 are particularly  simple for
 unfrustrated,
isotropic, homogeneous lattices with nearest neighbor hopping
(sect.\ \ref{sect:resapp}, appendix \ref{app:unfrustrated}).
For frustrated, non-bipartite lattices our approach is still
tractable, though more cumbersome. To demonstrate its
tractability we derived formulae for the triangular lattice
(non-bipartite, Bravais lattice) and for the kagome lattice
(non-bipartite, non-Bravais lattice).

We believe that our variational criteria are well suited in order to
investigate the lattice dependence of saturated ferromagnetism in the Hubbard
model since they cover the most relevant local excitations in the spin-up
Fermi sea but are still simple enough to be evaluated routinely
on various lattices
in various dimensions. Local and non-local band narrowing effects
are present in these approaches. Since this fact is not obvious in the
complete approaches we resort to the previous result (\ref{skaband}).
The factor $\delta(1-(e_1/\delta z t)^2)$ clearly describes the band
narrowing of the flipped spin. It comprises two factors one of which
is local ($\delta$) and thus survives also in the limit $d\to\infty$.
The other factor ($1-(e_1/\delta z t)^2)$ is very important as well since it
vanishes equally on $\delta\to0 $. But in the limit $z\to \infty$ on
scaling $t\propto 1/\sqrt{z}$ \cite{metzn89a} the latter factor
degenerates to unity. This clearly shows its non-local character.
 Note the importance of the sequence of limits.
The more sophisticated variational approaches discussed in the present
work (RES0 - RES3) comprise the ansatz (\ref{ansatz:gw}). Thus they
contain also local and non-local band narrowing effects.
The other main effect is a direct energy lifting of the minority
electron due to the infinite (or large) on-site repulsion. Since
the minority electron blocks a site, the majority electrons loose the
kinetic energy related to hopping onto or from this site, namely
$e_1$. This is seen best in the kinetic matrix elements
in (\ref{xelemb}) or in the energy denominator (\ref{fdef}).
Futhermore, we like to draw the reader's attention to the
non-orthogonality as it can be discerned in (\ref{xelema}).
It is very difficult to comprehend its effect intuitively.
But we know from the extensive efforts to reduce the critical
doping by including more and more correlations \cite{wurth96}
that this non-orthogonality hinders the spin flip to gain enough
energy to destabilize the Nagaoka state. The added states do not reduce
the critical doping any further since they do not really enhance the
accessible Hilbert space.

Besides the achievement of easily evaluated ansatzes the comparison
of the phase diagrams presented here yields the following  main results.
For bipartite lattices the possible Nagaoka region shrinks
rapidly with increasing coordination number $z$ (cf. square
and simple cubic lattice). Herrmann and Nolting did not
investigate low-dimensional lattices because they suppose that
ferromagnetism is excluded in $d=1$ and 2 by the
Mermin-Wagner theorem \cite{herrm97a,herrm97b}.
Note, however, that neither the  Mermin-Wagner theorem
 makes any statement on ground states \cite{mermi66}
nor any extended theorem can exclude
a ferromagnetic ground state since the total spin as \textit{conserved}
quantity is not affected by quantum fluctuations.
The shrinking of the Nagaoka region on increasing coordination number
 can be understood from the lowering of the DOS at the band edges or,
equivalently, as effect of a lower and lower band edge $\varepsilon_{\rm b}$.

For the above reasons we investigated low-dimensional non-bipartite lattices
where low DOS at the lower band edge can be avoided. Indeed, we found that the possible Nagaoka regions are enlarged
considerably. This is true for electron doping for conventional hopping
($t>0, n>1$) whereas saturated ferromagnetism in the hole doped region
($t>0, n<1$) can be excluded by our results.
Treating the electron doping also as hole doping after a particle-hole
transformation, i.e.\ $t>0, n>1$ $\to$ $t<0, n<1$, this phenomenon is
easily understood: $\varepsilon_{\rm b}(t<0) < \varepsilon_{\rm b}(t>0)$.
The ratio of the lower band edges is $2$ for the triangular and the
kagome lattice, and $3$ for the \textit{fcc} and \textit{hcp} lattice. In infinite
dimensions it becomes even $\infty$ for the generalizations of the \textit{fcc} lattice
\cite{mulle91,uhrig96a,ulmke96}. For these generalizations
one has
$\varepsilon_{\rm b}(t>0) / \varepsilon_{\rm b}(t<0) \propto \sqrt{d}$.

The above observations concern already the asymmetry of the density of
states. Our results clearly show that a large asymmetry favors ferromagnetism.
It is most useful to have a {\em large DOS at the lower band edge} in the hole
doping picture to stabilize the Nagaoka state. Note that this is {\em not}
equivalent to the well-known Stoner criterion $U\rho(\varepsilon_{\rm F})>1$
which concerns only the DOS at the Fermi level \cite{vollh97}.
 The best situation is to
have a strongly diverging singularity at the lower band edge or close to it
as we found
in the investigation of the $t$-$t'$ model with  tunable DOS and as was also
observed previously for \textit{fcc}-type lattices
\cite{uhrig96a,ulmke96,herrm97b}.

Our results concerning the $t$-$t'$ model extend previous
ones \cite{hlubi97} since we treat all ratios of $t$ and
 $t'$ and all fillings $n$. Hlubina
{\it et al}. focussed on the Fermi levels at the van-Hove singularity. Thus
the Stoner criterion is  at the basis of their investigation albeit it
goes technically beyond this mean-field criterion.

Herrmann and Nolting used a two-pole method (SDA: self-consistent spectral
density approach)
to investigate ferromagnetism
for the simple cubic, the $d=\infty$ hypercubic,
 the $d=\infty$ \textit{fcc}, and the \textit{bcc}
 lattice \cite{herrm97a,herrm97b}
at zero and at finite temperature.
Their qualitative findings for zero temperature are similar to ours.
We like, however, to point out that the two-pole method they employ
is indeed a generalization of the Gutzwiller ansatz in (\ref{ansatz:gw})
with $f=0$
to finite temperatures and non-saturated magnetizations.
 For $T=0$ and saturation it reduces to (\ref{ansatz:gw}) with $f=0$.
Thus it is not astounding that they found a good agreement to the results
of Shastry {\it et al.}\ \cite{shast90a}. Our approaches go far beyond
(\ref{ansatz:gw}) (barring the question of the
 extendability to finite temperatures). This can be seen for instance
 for the simple cubic lattice where we found $\delta_{\rm cr}=0.237$
well below  $\delta_{\rm cr}=0.32$ \cite{shast90a,herrm97b}. Already
Roth found by numerical calculation in the variational subspace of RES0
the number $\delta_{\rm cr}=0.24$ \cite{roth69}.
For the \textit{bcc} lattice one finds again that the SDA method
 \cite{herrm97b}
reproduces the Gutzwiller result $\delta_{\rm cr}=0.324$ for
saturated ferromagnetism whereas RES0 yields a considerably lower
value of $\delta_{\rm cr}=0.239$.
Thus one is led to the conclusion that the SDA two-pole method cannot
be exact as claimed  in the strong coupling limit \cite{herrm97a}.

\section*{Acknowledgements:}
This work was performed within the research program of the
Sonderforschungsbereich 341 supported by the Deutsche Forschungsgemeinschaft.
The authors gratefully acknowledge useful discussions with Peter Wurth and
Burkhard Kleine.


I answered this question on Quora, and thought it was worthwhile posting the answer here, as well.
Q: Is it easier to get over someone you cared for once by enthusiastically hating them?
Actually, while hating is not ultimately a rewarding or useful emotion, in the short term it could be a practical way to reframe the physiological state of broken-heartedness and longing for a lost love. Hate is related to love as a state in the body and brain.
Unfortunately, advice to just forget about him or her doesn't work well. It's difficult to force emotions to change, and it's not emotionally healthy. When you tamp down any emotion, even a negative one, you are reducing your ability to fully feel all emotions.
FMRI studies have shown that the state of romantic love is similar to reward-seeking and competitive activities, which usually are characterized by high levels of dopamine. (Dopamine is the brain chemical of pleasure, but levels begin to drop as soon as you get the reward.) It's likely that the brain chemistry of the person dumped is also high in testosterone (lusting for the beloved) and oxytocin (being bonded to the beloved).
A couple of recent studies have shown that oxytocin also plays a part in aggression toward outsiders; testosterone is also a chemical of aggression; and dopamine creates high focus on its object.
Biologists define emotion as a physiological state that we then name using the prefrontal cortex. So, if our broken-hearted person already has lots of oxytocin, testosterone and dopamine circulating in his or her brain, creating these responses, it should not be too difficult to reframe the emotion as hate. "He's awful. I wouldn't want her."
If you remove the reward-seeking element, dopamine levels should drop, allowing the focus on the other person to fade away.


Every day from 4pm to 6pm, we offer a list of treats to celebrate the end of the work day.
Every Monday night we offer a special burger, paired with a world class craft brew. The pairings are never the same, so check our social feeds to see what our kitchen team has crafted this week.
Each Friday we feature special cuts from our local farms. It could be a giant bone-in pork chop from Wild Harmony Farm, dry aged beef ribs or a 40 ounce 30-day dry aged bone-in rib eye from Beriah Lewis Farm, whatever we feature, it will be epic.
The usual suspects and more...Bloody Mary's, mimosas, breakfast beers and comfort food to cure you. Sunday from 10am to 2:30pm.


We have all experienced intense buzzing in our ears after a nightout in a club with loud music, especially the moment we go to sleep. This buzzing is called tinnitus. It is the first sign of a temporary hearing impairment. It has been proved that it may, either in short terms or directly as a hearing injury, lead to a permanent, painful and exhausting state, able to lead even in mental disorders. At the same time, this temporary injury may cause hearing loss. Soundz Uni Night Out earplugs are the best choice for your hearing protection during your night out.
The case contains 2 pairs of Soundz eartips so that you can choose those that fit better in your ears, 1 pair of Soundz NightOut filters, 1 Soundz Hard Zipper Case.
Soundz Uni Night Out earplugs can help you improve the quality of your life.


text:
Са́ндерс () — англійське, нідерландське або північнонімецьке патронімічне прізвище, у буквальному значенні — «син Александера». Прізвище походить від варіанту скорочення імені xander, яке пов'язане з грецьким «Ἀλέξανδρος» (Aléxandros), у значенні «захисник людей».

Відомі носії прізвища

 Берні Сандерс (;  1941) — американський політик, кандидат на праймеріз Демократичної партії в 2016.
  (;  1942) — американський письменник і художник.
 Волтер Джеремі Сандерс (;  1936) — американський бізнесмен.
 Гарланд Сандерс (; 1890—1980) — американський підприємець, засновник KFC.
 Джозеф Сандерс (; 1773—1853) — британський художник, гравер, ілюстратор, видавець і професор образотворчого мистецтва.
 Джордж Сандерс (; 1906—1972) — британський актор, володар премії «Оскар» (1951).
 Едвард Сандерс:
 Едвард Сандерс (; 1930—1954) — американський боксер, олімпійський чемпіон (1952).
  (;  1993) — британський актор і співак.
  (;  1995) — американський актор.
 Коррі Сандерс (; 1966—2012) — південноафриканський професійний боксер важковаговик.
 ; (;  1962) — американський аніматор, актор озвучення, сценарист, режисер і продюсер.
 Люк Сандерс (;  1945) — бельгійський футболіст, що грав на позиції воротаря.
 Патрік Сандерс (;  1966) — старший офіцер британської армії, начальник Генерального штабу з червня 2022 року.
 Рональд Сандерс (;  1948) — дипломат Антигуа і Барбуда.
  (;  1971) — британський кінорежисер.
 Саммер Сандерс (;  1972) — американська плавчиня, олімпійська чемпіонка (1992).
 Сара Гакабі Сандерс (;  1982) — американська політична діячка, член Республіканської партії.

Див. також
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