Magnetism and the Weiss Exchange Field - A Theoretical Analysis Inspired by Recent Experiments
C. Albert\footnote{supported in part by the Swiss National Foundation}
\
L. Ferrari\footnote{present address: Collegio Papio, Ascona, Switzerland}\
J. Fr\"ohlich\
Theoretical Physics
ETH-H\"onggerberg
CH-8093 Z\"urich
The huge spin precession frequency observed in recent experiments with spin-polarized beams of hot electrons shot through magnetized films is interpreted as being caused by Zeeman coupling of the electron spins to the so-called {\em Weiss exchange field} in the film. A ``Stern-Gerlach experiment'' for electrons moving through an inhomogeneous exchange field is proposed. The microscopic origin of exchange interactions and of large mean exchange fields, leading to different types of magnetic order, is elucidated. A microscopic derivation of the equations of motion of the Weiss exchange field is presented. Novel proofs of the existence of phase transitions in quantum $XY$-models and antiferromagnets, based on an analysis of the statistical distribution of the exchange field, are outlined.
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\begin{document}
\title{Magnetism and the Weiss Exchange Field - A Theoretical Analysis Inspired by Recent Experiments}
\author{C. Albert\footnote{supported in part by the Swiss National Foundation},\,\, L. Ferrari\footnote{present address: Collegio Papio, Ascona, Switzerland}\,\,, J. Fr\"ohlich\,\,, B. Schlein\footnote{present address: Department of Mathematics, Stanford University, Stanford CA
94305, USA}\\
Theoretical Physics, ETH-H\"onggerberg, CH-8093 Z\"urich}
\maketitle

\begin{abstract}
The huge spin precession frequency observed in
recent experiments with spin-polarized beams of hot electrons shot
through magnetized films is interpreted as being caused by Zeeman
coupling of the electron spins to the so-called {\em Weiss
exchange field} in the film. A ``Stern-Gerlach experiment'' for
electrons moving through an inhomogeneous exchange field is proposed. The
microscopic origin of exchange interactions and of large
mean exchange fields, leading to different types of
magnetic order, is elucidated. A microscopic derivation of the equations of motion of the Weiss
exchange field is presented. Novel proofs of the
existence of phase transitions in quantum $XY$-models and
antiferromagnets, based on an analysis of the statistical
distribution of the exchange field, are outlined.
\end{abstract}

\section{Introduction}
\setcounter{equation}{0}

Effects of ferromagnetism have been known since antiquity. But a mathematical understanding of the microscopic origin of ferromagnetism has remained somewhat elusive, until today! Pauli paramagnetism, ferro-, ferri- and antiferromagnetism are quantum phenomena connected to the spin of electrons and to Pauli's exclusion principle. The theory of {\em paramagnetism} in (free) electron gases is quite straightforward, \cite{A}.
{\em Antiferromagnetism} is relatively well understood: A mechanism for the generation of antiferromagnetic exchange interactions has been proposed by Anderson \cite{5}, who discovered a close relationship between the half-filled Hubbard model and the Heisenberg antiferromagnet using perturbative methods; (see also \cite{6} for mathematically more compelling and more general variants of Anderson's key observation). It has been proven rigorously by Dyson, Lieb and Simon \cite{4}, using the method of infrared bounds previously discovered in \cite{3}, that the {\em quantum Heisenberg antiferromagnet} with nearest-neighbour exchange couplings exhibits a phase transition accompanied by spontaneous symmetry breaking and the emergence of gapless spin waves, as the temperature is lowered, in {\em three} or {\em more} dimensions. (The Mermin-Wagner theorem says that, in (one and) two dimensions, continuous symmetries cannot be broken spontaneously in models with short-range interactions, \cite{B}.)

Our mathematical understanding of {\em ferromagnetism} is far
less advanced. Some kind of heuristic theory of
ferromagnetism emerged, long ago, in the classic works of
Heisenberg, Bloch, Stoner, Dyson, Landau and Lifshitz, and others
\cite{1}. Various insights have been gained on the basis of some
form of mean-field theory, with small fluctuations around
mean-field theory taken into account within a linear
approximation. This approximation, however, is known to break down in the
vicinity of the critical point of a ferromagnetic
material, where nonlinear fluctuations play a crucial role,
\cite{2}.

In a variety of tight-binding models of itinerant electrons,
ferromagnetic order has been exhibited in the {\em ground state}
(i.e., at zero temperature); see \cite{C},\cite{D},\cite{E}. One
of these models is a fairly natural two-band model in which
ferromagnetism arises from a competition between electron hopping,
Coulomb repulsion and an on-site Hund's rule, \cite{E}. (Hund's
rule says that the spin-tripled state of two electrons occupying
the same site is energetically favoured over the spin-singlet
state. It should be emphasized, however, that a mathematically
rigorous derivation of Hund's rule in atomic physics from first
principles has {\em not} been accomplished, so far.) None of the
results in \cite{C},\cite{D},\cite{E} comes close to providing
some understanding of ferromagnetic order and of an order-disorder
phase transition at {\em positive} temperature. It is not known
how to derive, with mathematical precision, an effective
Hamiltonian with explicit ferromagnetic exchange couplings from
the microscopic Schr\"odinger equation, or a tight-binding
approximation thereof, of ferromagnetic materials. But even if we
resort to a phenomenological description of such materials in
terms of models where ferromagnetic exchange couplings have been
put in {\em by hand} we face the problem that we are unable to
exhibit ferromagnetic order at low enough temperature and to
establish an order-disorder phase transition in three or more
dimensions. {\em No mathematically rigorous proof} of the phase
transition in the {\em quantum Heisenberg ferromagnet} is known,
to date! (Such a result has, however, been established for {\em
classical} Heisenberg models in \cite{3}.)

Ab-initio quantum Monte Carlo simulations of models of quantum
ferromagnets are plagued by the well known ``sign- (or
complex-phase) problem''.

Thus, until now, there are neither substantial mathematically rigorous results on, nor are there reliable ab-initio numerical simulations of, realistic models of ferromagnetic metals, such as Ni, Co or Fe! Given that ferromagnetism is among the most striking {\em macroscopic} manifestations - apparent, e.g., in the needle of a compass - of the {\em quantum-mechanical} nature of matter, this is clearly a desolate state of affairs.

In the present paper we shall not remedy this unsatisfactory situation. However, first, we attempt to draw renewed attention to it, and, second, we outline a formalism and some fairly elementary analytical observations of which we hope that they will ultimately lead to a better, mathematically rather precise understanding of ferromagnetism.
Were it not known already, our analysis and the one in \cite{E} would make clear that ferromagnetism is a non-perturbative phenomenon involving strong correlations and gapless modes. To understand it mathematically will most probably necessitate a full-fledged {\em multi-scale} (renormalization group) {\em analysis}. The formalism presented in this paper and our calculations are intended to provide a convenient starting point for such an analysis.

Analytical work on ferromagnetism may seem to be rather
unfashionable. However, there are recent developments, such as
spintronics, fast magnetic devices, etc. that may make work like
ours appear worthwhile. Our own motivation for the work that led
to this paper actually originated in studying recent experiments
with beams of spin-polarized, hot electrons shot through
ferromagnetically ordered films consisting of Ni, Co or Fe that
were carried out in the group of H.C. Siegmann at ETH; see
\cite{7},\cite{8}. Back in 1998, it became clear to one of us that
the concept of the ``{\em Weiss exchange field}'' (see
\cite{9},\cite{13}) would play a useful role in a theoretical
interpretation of the experimental results reported in
\cite{7},\cite{8}. More generally, the Weiss exchange field appears to
offer a key to a systematic study of phase transitions in magnetic
materials, magnetic order, spin precession and magnon dynamics. In this paper, we focus on elucidating the microscopic origin of the Weiss
exchange field, the role it plays in the theory of magnetism, and
its dynamics.

Our paper is organized as follows.

In Section 2, we describe the experiments reported in \cite{7},\cite{8} and sketch a phenomenological interpretation, based on scattering theory, of the results found in these experiments, merely adding some conceptual remarks to the discussion of our experimental colleagues and describing the role played by the Weiss exchange field. We also propose some further experiments, in particular a Stern-Gerlach experiment for electrons traversing an inhomogeneous exchange field.

A mathematically precise analysis of the scattering of electrons (or neutrons, photons, ...) at dynamical targets, such as magnetic films, metallic solids, liquid droplets, ..., will appear elsewhere; (some first results appear in \cite{F}).

In Section 3, we reformulate one-band $t-J$ and Hubbard models in terms of a dynamical Weiss exchange field. For this purpose, the standard imaginary-time functional integral formalism for the analysis of thermal equilibrium states of quantum many-body systems is recalled.
The {\em Weiss exchange field} is seen to be a Lagrange-multiplier field in a Hubbard-Stratonovich transformation of the original functional integral that renders the action functional quadratic in the Grassmann variables describing the electronic degrees of freedom; see e.g. \cite{13},\cite{11}. The effective field theory of the Weiss exchange field is obtained after integrating over those Grassmann variables.
The effective (imaginary-time) action functional of the exchange field and identities for Green functions of spin operators are derived.

In Section 4, we determine the leading terms of the effective action of the Weiss exchange field, ${\bf W}$, in the approximation where fluctuations of the {\em length} of the exchange field are neglected.
For this purpose, we derive the ferro- and antiferromagnetic mean-field equations from the exact effective action of the exchange field.
By solving these equations we determine the most likely length, $W_0$, of the exchange field. From that point on, the length of the exchange field is frozen to be $|\vc W|\equiv W_0$.

We then consider a one-band Hubbard model with a half-filled band and find that, in this situation, the effective action of $\vc W$ is the one of a nonlinear $\sigma$-model with a minimum that favours N\'eel order. This result is found on the basis of controlled perturbative calculations and goes beyond linear stability analysis of the antiferromagnetic mean-field solution, (which has been presented, e.g., in \cite{13}).
It represents a functional-integral version of Anderson's basic observations \cite{5}.

We then turn to ferromagnetically ordered mean field solutions and show that $x$-independent fluctuations are not a source of instability of such a solution.
Then we consider a one-band Hubbard model with a weakly filled, fairly flat band. In this situation one expects that ferromagnetism prevails. Indeed, we find that, at low temperatures, the ferromagnetic mean-field equation has a non-trivial solution, and that this solution belongs to a {\em quadratically stable} critical point of the effective action of {\vc W}. This conclusion is the result of somewhat subtle calculations involving processes close to the Fermi surface, which make the dominant contribution (but would lead to small-energy denominators in a purely perturbative analysis). Details will appear in \cite{G}.
Our calculations support the idea that the one-band Hubbard model with a weakly filled, fairly flat band describes coexistence of metallic behaviour with ferromagnetic order, at sufficiently low temperatures. A similar conclusion was reached, tentatively, in \cite{E} for some two-band Hund-Hubbard models. The methods of the present paper also apply to the model discussed in \cite{E}; see \cite{G}.

In the next to last subsection of Section 4, we exhibit a universal
Wess-Zumino term in the effective action of the Weiss exchange
field $\vc W$ and calculate its coefficient, which is purely
imaginary. The Wess-Zumino term is ``irrelevant'' for
antiferromagnets, but plays a {\em crucial} role in the dynamics
of magnons in {\em ferromagnetically} ordered systems. Repeating
arguments in \cite{17}, we derive the Landau-Lifshitz equations
for magnons in a ferromagnet.

Finally, we draw attention to two well known arguments explaining why there is no magnetic ordering at positive temperature, in one and two dimensions; (but see \cite{B}).

In Section 5, we sketch novel rigorous proofs, based on analyzing the effective field theory of the exchange field, $\vc W$, of the existence of phase transitions and magnetic order at low temperatures in a class of $XY$-models, Heisenberg antiferromagnets and ferromagnets of localized $SO(2n)$- spins, for $n=1,2,\dots$. See \cite{4} for the original results.
Our proof is based on establishing reflection positivity of the effective field theory of the exchange field $\vc W$ and then using the original techniques developed in \cite{3}; (see also \cite{14}, \cite{14II}).

It should be emphasized that the concept of the Weiss exchange field has a number of further, quite exciting applications. We hope to return to these matters in future papers.

Some of the material in this paper has a review character; but
some of it is new. We hope it is fairly easy to read. If it draws
renewed attention to some of the deep technical problems in the
quantum theory of magnetism it has fulfilled its purpose. We
gratefully dedicate this paper, belatedly, to two great colleagues
and friends of the senior author (J.F.): G. Jona-Lasinio, on the
occasion of his seventieth birthday, and H.-C. Siegmann, on the
occasion of his retirement from ETH.

{\it Acknowledgements.} J. Fr\"ohlich thanks S. Riesen and H.-C.
Siegmann for very stimulating discussions of the
experiments in \cite{7}, P. Wiegmann for some crucial advice with the
calculations in Section 4, and the IH\'ES for
hospitality during much of the work on this paper.
We all thank M. Azam for very useful and pleasant discussions on the uses of the exchange field.

\section{Real and gedanken experiments involving the Weiss
exchange field} \label{sec:exp} \setcounter{equation}{0}

We start this section with a brief description of recent
experiments carried out by Oberli, Burgermeister, Riesen, Weber
and Siegmann at ETH-Z\"urich \cite{7}, \cite{8}. In these experiments, a beam of hot,
spin--polarized electrons is shot through a thin ferromagnetic film (Ni,
Co, or Fe) and the polarization of the outgoing beam is observed.
Their experimental setup is as described in Fig. \protect\ref{fig:exp1}.

\begin{figure}[h]
\begin{center}
\input{exp.pstex_t}
\end{center}
\caption{Experimental setup}\label{fig:exp1}
\end{figure}


The following quantities are measurable:
\begin{itemize}
\item[i)] The thickness, $d$, of the film; $d$ is a few
nanometers. \item[ii)] The average energy, $E$, of an incident
electron; if $E_F$ denotes the Fermi energy of the magnetic film
then $E -E_F$ varies between 4$eV$ and 16$eV$. The group velocity
of the electrons inside the film is denoted by ${\bf v}$; it is
not directly measurable, but it is comparable to
$(2(E+eV)/m_{\text{el}}^*)^{1/2}$, where $eV$ is the average potential energy of an electron and $m_{\text{el}}^*$ its effective mass inside the film.
\item[iii)] The degree, $P_0$, and the direction, ${\bf n}_0$, of
the spin polarization of the incident electron beam; (in Fig.
\ref{fig:exp1}, ${\bf n}_0$ is parallel to the $x$-direction, ${
\bf v}$ to the $y$-direction); the same quantities, $P$ and ${ \bf
n}$, for the outgoing beam. \item[iv)] The direction of the
magnetization, ${ \bf M}$, of the film (in Fig. \ref{fig:exp1}
chosen to be parallel to the $z$-axis); the angles, $\theta_0$ and
$\theta$, between ${ \bf n}_0$ and ${ \bf M}$ and between ${ \bf
n}$ and $\bf M$, respectively ($\theta_0 = \pi /2$, in Fig.
\ref{fig:exp1}). Experimentally, the angle $\theta$ is found to be
considerably smaller than $\theta_0$, i.e., the spins of the
transmitted electrons rotate into the direction of the spontaneous
magnetization ${ \bf M}$ of the film. This is interpreted as being
mainly due to an enhanced absorption of minority-spin electrons,
as compared to majority-spin electrons; see (vi). (It appears that
the contribution of spin flip processes accompanied by magnon
emission into the film -- ``Stoner excitations'' -- to the total
spin rotation is only around 5\%, \cite{7}). \item[v)] The spin
precession angle, $\epsilon$, between the projections of ${ \bf
n}_0$ and of ${ \bf n}$ onto the plane perpendicular to ${ \bf M}$
(the $xy$-plane of Fig. \ref{fig:exp1}); $\epsilon$ is found to be
``large'' (tens of degrees). \item[vi)] Let $I$ be the intensity
of the incident beam, and let $I^+ = I^+ (E)$ and $I^- = I^- (E)$
be the intensities of the outgoing beam of electrons with spin
parallel or antiparallel to ${ \bf M}$, respectively, assuming the
incident beam has intensity $I$ and the spins of its electrons are
parallel to ${ \bf M}$ (${ \bf n}_0$ parallel to ${ \bf M}$, $P_0
\cong 1$), or antiparallel to ${ \bf M}$ (${ \bf n}_0$
anti-parallel to ${ \bf M}$, $P_0 \cong 1$), respectively. Then
$\theta_0 = \theta =0$, or $\theta_0 = \theta =\pi$, respectively,
and $\epsilon =0$. $I^+$ and $I^-$ can be measured and yield the
spin-transmission asymmetry \[ A = \frac{I^+ - I^-}{I^+ + I^-} ;\]
$A$ is found to be positive and large. This is interpreted in
terms of rates of transitions of electrons into unoccupied 3d
states (holes): There are more unoccupied 3d states in the film
with spin antiparallel to ${ \bf M}$ (minority spin) than with
spin parallel to ${ \bf M}$ (majority spin). This explains
qualitatively the experimental results found for $A$ and for
$\theta - \theta_0$ (see (iv)); \cite{7}. \item[vii)] The orbital
deflection angle, $\alpha$, between the directions of the incident
and the transmitted beam (not indicated in Fig. \ref{fig:exp1}).
Experimentally $\alpha$ is found to be negligibly small. This
tells us that the integrated Lorentz force on the electrons
transmitted through the film is tiny. The precession of the spins
of the electrons when they traverse the film can therefore {\it
not} be explained by Zeeman coupling of the spins to the magnetic
field inside the layer. It is mainly due to Zeeman coupling of the
spins to what will be called the {\it Weiss exchange field}. In
iron, the Weiss exchange field causing the observed spin
precession would correspond to a magnetic field of roughly 8000
Tesla (which is gigantic).
\end{itemize}

A theoretical interpretation of the experimental results reported
in \cite{7} can be attempted within the formalism of {\it
scattering theory}. If the luminosity of the incident beam is low
we can consider a single incoming electron. The incoming state is
described as a tensor product of a Pauli spinor,
$\psi_{\text{in}}$, describing the incident electron and a state,
$\xi$, of the film. Typically, $\xi$ is the ground state
(temperature $T=0$) or a thermal equilibrium state ($T >0$) of the
film. The outcoming state, long after the interactions between an
outgoing electron and the film have taken place, is more
complicated and will, in general, exhibit {\it entanglement}
between the electron and the degrees of freedom of the film. If
only measurements far away from the film are performed, as in
\cite{7}, the outgoing state can be described as a density matrix
\begin{equation}
P_{\text{out}} = (\rho_N)_{N=0}^{\infty},
\end{equation}
where $\rho_N$ is a non-negative, trace-class operator on the
Hilbert space of $N$ {\it outgoing electrons} (the incident
electron has knocked $N-1$ electrons out of the film), $N
=2,3,\dots$; $\rho_0$ is a non-negative number, the {\it
absorption probability}, $\rho_1$ is a non-negative trace-class
operator on the Hilbert space
\begin{equation}
\mathcal{H} = L^2 (\bR^3) \otimes \bC^2
\end{equation}
of square-integrable Pauli spinors and describes the (generally
{\it mixed}) state of {\it one} outgoing electron. ``Conservation
of probability'' implies that
\begin{equation}\label{eq:cons}
\rho_0 + \sum_{N=1}^{\infty} \tr \rho_N =1 .
\end{equation}
The state $P_{\text{out}}$ is obtained by taking a partial trace
of the outgoing state of the {\it total system, including} the
film, over the degrees of freedom of the film. This is justified,
because the degrees of freedom of the film are {\it not} observed
in the experiment. If the energy, $E$, of the incident electron is
below (or comparable to) the threshold, $\Sigma^{(2)}$, for
emission of two or more electrons from the film then
\begin{equation}\label{eq:rhoN}
\rho_N = 0 \quad \text{for } N \geq 2.
\end{equation}
In the interpretation of the experimental data provided in
\cite{7}, this is tacitly assumed.

Experimentally, the absorption probability $\rho_0$ and the spin
polarizations ${ \bf P}_0$ and ${ \bf P}$ of the incoming and the
outgoing electron, respectively, are measured. The vectors ${ \bf
P}_0$ and ${ \bf P}$ are given by
\begin{equation}
{ \bf P}_0 \equiv P_0 { \bf n}_0 = \langle \psi_{\text{in}} , {
\pmb \sigma} \psi_{\text{in}} \rangle ,
\end{equation}
and
\begin{equation}
{ \bf P} \equiv P { \bf n} = \frac{1}{ \tr_{\cH} (\rho_1)} \,
\tr_{\cH} (\rho_1 { \pmb \sigma}) ,
\end{equation}
where $\psi_{\text{in}}$ is the wave function of the incident
electron, ${ \pmb \sigma} = (\sigma_x, \sigma_y,\sigma_z)$ is the
vector of Pauli matrices, and $\widetilde \rho_1 := [\tr_{\cH}
(\rho_1)]^{-1} \rho_1$ is the {\it conditional} state of the
outgoing electron, given that it has {\it not} been absorbed in
the film. If (\ref{eq:rhoN}) is assumed to hold then
\begin{equation}
\tr_{\cH} (\rho_1) = 1 -\rho_0
\end{equation}

Since we have assumed that the incoming electron has been prepared
in a pure state,
\begin{equation}
P_0 \equiv |{ \bf P}_0|=1 .
\end{equation}
However, there is, a priori, no reason why $\widetilde \rho_1$
should be a pure state. If it were pure then
\begin{equation}
P = |{ \bf P}|=1 .
\end{equation}

It would be highly interesting to estimate, experimentally, the
amount of entanglement with the film (or {\it decoherence}) in the
state of the outgoing electron by measuring the quotient $P/P_0$.
If $P /P_0 <1$ then $\widetilde \rho_1 = [\tr_{\cH} (\rho_1)]^{-1}
\rho_1$ is {\it not} a pure state, anymore, meaning there is
entanglement with the film. Apparently, $P/P_0$ has not been
measured accurately, yet.

A moment's reflection shows that spin flip processes accompanied
by magnon emission in the film (``Stoner excitations'') lead to
entanglement; while absorption of electrons into unoccupied 3d
states need not be correlated with entanglement of the states of
{\it those} electrons that {\it do} traverse the film. In fact, it
is implicitly assumed in \cite{7} that if Stoner excitations are
neglected then $\widetilde \rho_1$ is close to a pure state (at
least in spin-space). The experimental techniques of \cite{7}
could be used to test this hypothesis.

Next, we express the spin transmission asymmetry $A$ (see (vi)) in
terms of outgoing states. Let $\psi^+_{\text{in}}$ and
$\psi^-_{\text{in}}$ be incoming states with spin polarization ${
\bf P}_0 \equiv { \bf P}_0^+$ parallel to ${ \bf M}$ (majority
spin) and ${ \bf P}_0 \equiv { \bf P}_0^-$ antiparallel to ${ \bf
M}$ (minority spin). Let $P^+_{\text{out}} =
(\rho_N^+)_{N=0}^{\infty}$ and $P_{\text{out}}^- =
(\rho_N^-)_{N=0}^{\infty}$ be outgoing states corresponding to
$\psi_{\text{in}}^+$, $\psi_{\text{in}}^-$, respectively. Then
\begin{equation}\label{eq:Asy}
 A= \frac{\rho_0^- - \rho_0^+}{2 - \rho_0^+ - \rho_0^-},
\end{equation}
as follows from (\ref{eq:cons}). More interesting would be
measurements of
\begin{equation}
{ \bf P}^{\pm} = \tr (\widetilde \rho_1^{\pm} { \pmb \sigma}).
\end{equation}
Clearly ${ \bf P}^{\pm}$ are parallel or anti-parallel to ${ \bf
M}$; but their lengths $P^{\pm} = | { \bf P}^{\pm}|$ ought to be
measured. In \cite{7}, it is tacitly assumed that $P^{\pm} \cong
1$, and that the states $\widetilde \rho_1^{\pm}$ are close to
{\it pure} states; but serious experimental data backing up this
hypothesis appear to be lacking. It is clear that it would be
invalidated if ``Stoner excitations'' played an important role.

In the following, we outline a phenomenological description of the
experiments in \cite{7}, assuming that (\ref{eq:rhoN}) and the
hypothesis just discussed (purity of $\widetilde \rho_1^{\pm}$)
are valid. (A more detailed discussion of the scattering approach
to electron transmission-- and reflection experiments will be
presented elsewhere.)

When an incoming electron enters the film it occupies an empty
state of the film. If the film is crystalline this state belongs
to a band of states; let $\alpha$ be the corresponding band index.
The state of an electron in band $\alpha$ is described by a Pauli
spinor
\begin{equation}
\phi = (\phi^+,\phi^-) ,
\end{equation}
where $\phi^+$ and $\phi^-$ are the components of $\phi$ with spin
parallel or anti-parallel to the magnetization ${ \bf M}$,
respectively. Adopting the approximation of the Peierls
substitution, the Pauli equation for $\phi$ in configuration space
has the form
\begin{equation}\label{eq:cD0}
i \hbar \cD_0 \phi = \mathcal{E}_{\alpha} (-i\hbar { \bf \cD})
\phi,
\end{equation}
where $\mathcal{E}_{\alpha} ({ \bf p})$ is the band function of
band $\alpha$,
\begin{align}
\cD_0 &= \frac{\partial}{\partial t} + i \frac{e}{\hbar} \phi^c +
i \left({ \bf W}_0^c \cdot \frac{{ \pmb \sigma}}{2}\right), \label{eq:cD-0}\\
\cD_j &= \frac{\partial}{\partial x^j} + i \frac{e}{\hbar} A_j + i
\left({ \bf W}_j \cdot \frac{{ \pmb \sigma}}{2}\right).
\end{align}
Here $\phi^c$ denotes a (complex) electrostatic potential, ${ \bf
W}_0^c$ is a (complex) {\it Weiss exchange field}, $(A_j) =
(A_1,A_2,A_3)$ is the electromagnetic vector potential, and $({
\bf W}_j) = ({ \bf W}_1 , { \bf W}_2, { \bf W}_3)$ is an
$SU(2)$-vector potential responsible for {\it spin-orbit
interactions}. As discussed in \cite{9}, (\ref{eq:cD0}) displays
electromagnetic $U(1)$-{\it gauge invariance and} $SU(2)$-{\it
gauge invariance}, i.e., covariance with respect to local
$SU(2)$-rotations in spin space. A number of important
consequences of these gauge symmetries have been pointed out in
\cite{9}.

According to (vii) above, effects of the electromagnetic vector
potential ${ \bf A}$ are apparently negligible; so ${ \bf A}$ is
set to 0. The electrostatic vector potential $\phi^c$ is given,
approximately, by
\begin{equation}
\phi^c = V + i\hat v
\end{equation}
where $eV$ is the surface exit work, and the imaginary part, $\hat v$,
of $\phi^c$ provides a phenomenological description of
spin-independent inelastic absorption processes inside the film.
Velocity-dependent spin-flip processes due to {\it spin-orbit
interactions} appear to play a very minor role in the experiments
reported in \cite{7}; so we may set ${ \bf W}_j$ to 0, $j=1,2,3$.

The Weiss exchange field ${ \bf W}_0^c$ is given by
\begin{equation}\label{eq:W0c}
{ \bf W}_0^c = { \bf W} -i{ \bf w},
\end{equation}
where the real part ${ \bf W}$ describes {\it exchange
interactions} between the incoming electron and the electron
density of the film, and the imaginary part, ${ \bf w}$, yields a
phenomenological description of spin-dependent absorption
processes.

Let $\phi_{\text{in}}$ denote the Pauli spinor describing the
state of an electron when it enters the film at some time $t_0$.
Eq. (\ref{eq:cD0}) can be solved for $\phi =\phi_t$, $t \geq t_0$,
with $\phi_{t_0} = \phi_{\text{in}}$. The solution is explicit if
${ \bf A}=0$, ${ \bf W}_j = 0$, $j=1,2,3$. Let us suppose that the
real part ${ \bf W}$ and the imaginary part ${ \bf w}$ of the
exchange field ${ \bf W}_0^c$ are both anti-parallel to the
magnetization ${ \bf M}$ of the film. Then (\ref{eq:cD0}) leads us
to consider two simple, quasi-one-dimensional scattering problems
in a complex potential well of depth
\begin{equation}
eV + ie\hat v \pm \frac{1}{2} (\Omega -i\omega),
\end{equation}
with $\Omega = |{ \bf W}|$ and $\omega =|{ \bf w}|$, for electrons with
spin parallel to ${ \bf M} (+)$, or anti-parallel to ${ \bf M}
(-)$, respectively. The solution to these scattering problems can
be found in every book on elementary quantum mechanics. For the
purposes of interpreting the results in \cite{7}, a semi-classical
treatment appears to be adequate. The group velocity, ${ \bf
v}_{\pm}$, of an incoming electron wave with energy peaked at $E$
and spin up $(+)$, or down $(-)$, inside the film can be found by
solving the equation
\begin{equation}\label{eq:xip}
\mathcal{E}_{\alpha} ({ \bf p}) = E-eV \mp \frac{1}{2} \Omega
\end{equation}
for ${ \bf p}$ and then setting
\begin{equation}
{ \bf v}_{\pm} = \frac{\partial \mathcal{E}_{\alpha}}{\partial {
\bf p}} ( { \bf p}^{\pm}),
\end{equation}
where ${ \bf p}^{\pm}$ is a solution of (\ref{eq:xip}) chosen such
that ${ \bf v}_{\pm}$ points in the positive $y$-direction. The
sojourn time $\tau_{\pm}$ of the wave inside the film is then
given by
\begin{equation}
\tau_{\pm} = \frac{d}{v_{\pm}}, \quad \text{with} \quad v_{\pm} =
|{ \bf v}_{\pm}|.
\end{equation}
If $\phi_{\text{in}}^{\pm}$ is the state of the electron with spin
up $(+)$, or spin down $(-)$, respectively, when it enters the
film its state $\phi_{\text{out}}^{\pm}$ upon leaving the film is
then given, approximately, by \footnote{Here and henceforth, we
use units such that $\hbar =1$.}
\begin{equation}\label{eq:phiout}
\phi_{\text{out}}^{\pm} \cong \exp \left( -i\tau_{\pm} [ (E-eV)
-ie\hat v \mp \frac{1}{2} (\Omega -i\omega)]\right) \phi_{\text{in}}^{\pm} .
\end{equation}
The presence of inelastic absorption processes implies that $e\hat v
\mp \omega/2 >0$. Setting
\begin{equation}
\exp \left( - \tau_{\pm} [\hat v \mp \omega/2]\right) = c \sqrt{1 \pm A},
\end{equation}
we find that
\begin{equation}\label{eq:phiout2}
\phi^{\pm}_{\text{out}} = c \sqrt{1 \pm A} e^{i(\theta \pm
\epsilon /2)} \phi^{\pm}_{\text{in}},
\end{equation}
where $A$ is the spin transmission asymmetry (see point (iv) and
(\ref{eq:Asy})), $\epsilon$ is the spin precession angle, and
$\theta$ is an (unimportant) spin-independent phase. If $v_+ \cong
v_- = v$ then $\tau_+ \cong \tau_- \cong d/v$, and
(\ref{eq:phiout}) yields
\begin{equation}
\epsilon \cong \tau_{\pm} \Omega \cong (d/v) |{ \bf W}| .
\end{equation}
Thus, measuring $\epsilon$ and $d$ and estimating $v$ yields an
approximate value for the size of the spin precession angular
velocity $\Omega$ and hence of the size of the Weiss exchange
field.

Eqs. (\ref{eq:phiout}) and (\ref{eq:phiout2}) could be mistaken
for equations describing {\it kaon}- or {\it neutrino
oscillations} and are analogous to the equations describing the
{\it Faraday rotation} of light traversing a magnetized medium.

The origin of the Weiss exchange field ${ \bf W}$ is hardly a
mystery: The spin of an electron traversing the film in a band
$\alpha$ apparently experiences (exchange) interactions with the spins
of the occupied states of the film. Since states with spin up and
with spin down are occupied asymmetrically (corresponding to the
fact that ${ \bf M} \neq 0$), the net spin density, ${ \bf S}
(x)$, at a point $x$ in the film is different from zero. The Weiss
exchange field ${ \bf W} = { \bf W} (x)$ is given by
\begin{equation}\label{eq:Wx}
{ \bf W} (x) = - J_{\alpha} { \bf S} (x),
\end{equation}
where $J_{\alpha}$ is the strength of the {\em exchange coupling}
between spins in the $\alpha^{\text{th}}$ band and those in the
occupied band. The theoretical discussion above is based on a {\it
mean field ansatz}: The exchange field ${ \bf W}$ in
(\ref{eq:cD-0}), (\ref{eq:W0c}) is chosen to be
\begin{equation}
{ \bf W} = \langle { \bf W} (x) \rangle = - J_{\alpha} \langle {
\bf S} (x) \rangle.
\end{equation}
(Our conventions are such that $\langle { \bf S} (x) \rangle$ is
parallel to the magnetization ${ \bf M}$.) One of the surprising
implications of the experimental results of \cite{7} is that,
apparently, $J_{\alpha}$ is quite large, even for rather
high-lying bands ($\alpha$), implying that the orbitals of states in such
bands must have substantial overlap with those of states in the
partially occupied, spin-polarized band.

Eq. (\ref{eq:Wx}) makes it clear that ${ \bf W} (x)$ is a {\it
dynamical} field. One of the main purposes of this paper is to
derive the effective quantum dynamics of ${ \bf W} (x)$ within a
Lagrangian functional integral formalism and to sketch what can be
accomplished with this formalism.

Before proceeding with this program, we outline some gedanken
experiments in the spirit of the experiments in \cite{7}.
\begin{itemize}
\item[i)] By applying an external magnetic field to the film
rotating in the $xy$-plane with angular velocity $\omega_0$, the
exchange field ${ \bf W}$ could be made to rotate around the
$z$-axis:
\begin{equation}
{ \bf W} (t) = W \left( \epsilon \cos (\omega_0 t + \delta) ,
\epsilon \sin (\omega_0 t + \delta) , \sqrt{1-\epsilon^2} \right).
\end{equation}
A polarized beam of electrons shot through this film must exhibit
{\it Bloch spin resonance}; but, in this experiment, it would be
due to the rotation of the exchange field. \item[ii)] One may
envisage a {\it Stern Gerlach experiment} for electrons. One would
start by constructing a sandwich of two ferromagnetic metals, I
(e.g. Fe) and II (e.g. Ni), with exchange fields ${ \bf W}_{I}$
and ${ \bf W}_{II}$ of different strength, joined by a transition
region, a mixture of I and II, of width $d_0$. The transition
region would be parallel to the $xy$-plane (see Fig.
\ref{fig:exp2}). One shoots an unpolarized beam of (not very hot)
electrons through the sandwich along the transition region between
I and II, as shown in Fig. \ref{fig:exp2}. One would expect to
detect two beams emerging on the other side of the film in
slightly different directions that are spin-polarized in opposite
directions.

\begin{figure}[h]
\begin{center}
\input{figure2.pstex_t}
\end{center}
\caption{Stern-Gerlach Experiment}\label{fig:exp2}
\end{figure}


The force, $f$, in the $z$-direction on an electron
with spin up/down inside the film is given, approximately, by
\begin{equation}
f \cong \pm \frac{|{ \bf W}_I - { \bf W}_{II}|}{2 d_0}.
\end{equation}
It yields a change in the $z$-component of the momentum of the
electron during its passage through the film given by
\begin{equation}
\Delta p_z \cong \pm \frac{|{ \bf W}_I - { \bf W}_{II}|}{2 d_0} \,
\frac{d}{v} ,
\end{equation}
where $v$ is the average group velocity. The deflection angle
$\alpha$ is found from
\begin{equation}
\tan \alpha = \frac{\Delta p_z}{\Delta p_y} .
\end{equation}
Of course, as discussed above, the intensity, $I_{+}$ of the upper
beam can be expected to be much larger than the intensity, $I_-$,
of the lower beam, due to spin-asymmetric absorption inside the
film. In order to achieve adequate focussing, the incident beam
could come from the tip of a scanning tunnelling microscope.
\item[iii)] In an experimental set-up similar to the one above,
one would force an electron current, spin-polarized in the
$z$-direction, through the film. Then a Hall tension in the
$z$-direction should be observed (Hall effect for spin currents;
see \cite{9}).
\end{itemize}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\input chapter3neu.tex2 \label{chap3}
\input chapter4.tex2
\input chapter5.tex2

\label{sec:reflpos}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{thebibliography}{100}
\bibliographystyle{unsrt}

\bibitem{A}
W. Pauli, {\em \"Uber Gasentartung und Paramagnetismus},
Z. Physik {\bf 41}, 81 (1927).

\bibitem{5}
P.W. Anderson,
{\em New approach to the theory of superexchange interactions},
Phys. Rev. {\bf 115}, 2-13 (1959).

\bibitem{6}
N. Datta, R. Fern\'andez, J. Fr\"ohlich,
{\em Effective Hamiltonians and phase diagrams for tight-binding models,}
J. Stat. Phys. {\bf 96}, 545-611 (1999).

\bibitem{4}
 F. Dyson, E. H. Lieb and B. Simon,
{\em Phase transitions in quantum spin systems with isotropic and non-isotropic interactions,}
J. Stat. Phys.  {\bf 18}, 335 (1978).

\bibitem{3}
J.~Fr\"ohlich, B.~Simon and T.~Spencer,
{\em Infrared Bounds, Phase Transitions And Continuous Symmetry Breaking,}
Commun.\ Math.\ Phys.\  {\bf 50}, 79 (1976).


\bibitem{B}
N.D. Mermin, H. Wagner, {\em Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models},
Phys. Rev. Lett. {\bf 17}, 1133-36 (1966).


\bibitem{1}
W. Heisenberg, Z. Physik {\bf 49}, 619 (1928).

F. Bloch,
Z. Physik {\bf 61}, 206 (1930).

E.C. Stoner, Proc. R. Soc. {\bf A165}, 372 (1938).

F.J. Dyson,
{\em General Theory of Spin-Wave Interactions,}
Phys. Rev. {\bf 102}, 1217-1230 (1956).

F.J. Dyson,
{\em Thermodynamic Behavior of an Ideal Ferromagnet,}
Phys. Rev. {\bf 102}, 1230-1244 (1956).

L.D. Landau, E.M. Lifshitz,
{\em Course of Theoretical Physics}
Volume 9, {\em Statistical Physics}, Part 2, by E.M. Lifshitz and L.P. Pitajewski, London: Pergamon 1981.

\bibitem{2}
E.~Brezin, J.~C.~Le Guillou and J.~Zinn-Justin,
{\em Field Theoretical Approach To Critical Phenomena,}
in
{\em Phase Transitions and Critical Phenomena,}  C. Domb and M.S. Green (eds.) Volume 6, 125-247, London: Academic Press 1976.

D.J. Amit,
{\em Field Theory, the Renormalization Group and Critical Phenomena},
New York a.o.: McGraw-Hill 1978.


\bibitem{C}
A. Mielke, H. Tasaki, {\em Ferromagnetism in the Hubbard model},
Commun. Math. Phys. {\bf 158}, 341-71 (1993).

\bibitem{D}
H. Tasaki, {\em Ferromagnetism in the Hubbard model: A constructive approach},
Commun. Math. Phys. {\bf 242}, 445-72  (2003).

\bibitem{E}
J. Fr\"ohlich, D. Ueltschi, {\em Hund's Rule and Metallic Ferromagnetism},
J. Stat. Phys.  {\bf 118}, 973-996 (2005).


\bibitem{7}
W.~Weber, S.~Riesen and H.~C.~Siegmann,
{\em Magnetization Precession by Hot Spin Injection},
Science {\bf 291}, 1015, (2001).

\bibitem{8}
D. Oberli,
{\em The ferromagnetic spin filter,}
PhD thesis ETH Z\"urich, Nr. 12933, (1998).
% http://e-collection.ethbib.ethz.ch/show?type=diss&nr=12933.

\bibitem{9}
J.~Fr\"ohlich, U.~M.~Studer and E.~Thiran,
{\em Quantum theory of large systems of nonrelativistic matter}, in
%Les Houches Summer School, Session 62: Fluctuating Geometries in Statistical Mechanics and Field Theory,
{\em Fluctuating geometries in statistical mechanics and field theory} (Les Houches 1994), F. David, P. Ginsparg, J. Zinn-Justin (eds.); Elsevier 1996,
[arXiv:cond-mat/9508062].

\bibitem{13}
E. Fradkin,
{\em Field theories of condensed matter systems},
Redwood City, California a.o.: Addison-Wesley 1994.

\bibitem{F}
J. Reuteler, {\em QM particles scattering off a dynamical target, and decoherence}, ETH-diploma thesis, spring 2005.

\bibitem{11}
T.~Chen, J.~Fr\"ohlich and M.~Seifert,
{\em Renormalization group methods: Landau-Fermi liquid and BCS superconductor,} in
{\em Fluctuating geometries in statistical mechanics and field theory}, F. David, P. Ginsparg, J. Zinn-Justin (eds.); Elsevier 1996,
[arXiv:cond-mat/9508063].

\bibitem{G}
M. Azam, L. Ferrari, J. Fr\"ohlich, to appear.

\bibitem{17}
H.~Leutwyler,
{\em Nonrelativistic effective Lagrangians,}
Phys.\ Rev.\ D {\bf 49}  3033 (1994),
[arXiv:hep-ph/9311264].


\bibitem{14}
J. Fr\"ohlich, R. Israel, E.H. Lieb, B. Simon,
{\em Phase Transitions and Reflection Positivity I. General Theory and Long Range Lattice Models,}
Commun. Math. Phys. {\bf 62}, 1-34, (1978).

\bibitem{14II}
J. Fr\"ohlich,
{\em Mathematical Physics of Phase Transitions and Symmetry Breaking},
Bulletin of the American Mathematical Society, {\bf 84}, 165-193 (1978).

\bibitem{10}
D. Ruelle,
{\em Statistical mechanics: rigorous results,}
Singapore: World Scientific, 1999.



\bibitem{12}
L. Birke, J. Fr\"ohlich,
{\em KMS, ETC.}
Rev. Math. Phys. {\bf 14}, 829-872, (2002),
[arXiv: math-ph/0204023].



\bibitem{15}
J.~Fr\"ohlich and T.~Spencer,
{\em Phase transitions in statistical mechanics and quantum field theory,}
in {\em New Developments in Quantum Field Theory and Statistical Mechanics}; M. L\'evy, P. Mitter (eds.), New York, London: Plenum 1977.

\bibitem{16}
E.H. Lieb,
{\em The Classical Limit of Quantum Spin Systems},
Commun. Math. Phys. {\bf 31}, 327-340, (1973).

\bibitem{18}
J.~Derezinski, V.~Jaksic and C.~Pillet,
{\em Perturbation theory of W*-dynamics, Liouvilleans and KMS-states,}
Rev.\ Math.\ Phys.\  {\bf 15} 447, (2003).

V.~Bach, J.~Fr\"ohlich and I.~M.~Sigal,
{\em Return to equilibrium,}
J.\ Math.\ Phys.\  {\bf 41} 3985 (2000) .

J. Fr\"ohlich, M. Merkli,
{\em Another Return of 'Return to Equilibrium'},
Commun. Math. Phys. {\bf 251}, 235-262, (2004).

\bibitem{Bach}
V. Bach, E. H. Lieb, J.P. Solovej, {\em Generalized Hartree-Fock Theory and the Hubbard Model},
[arXiv:cond-mat/9312044].

\end{thebibliography}
\end{document}

cond-mat0504674.extracted_bib
\begin{thebibliography}{100}
\bibliographystyle{unsrt}

\bibitem{A}
W. Pauli, {\em \"Uber Gasentartung und Paramagnetismus},
Z. Physik {\bf 41}, 81 (1927).

\bibitem{5}
P.W. Anderson,
{\em New approach to the theory of superexchange interactions},
Phys. Rev. {\bf 115}, 2-13 (1959).

\bibitem{6}
N. Datta, R. Fern\'andez, J. Fr\"ohlich,
{\em Effective Hamiltonians and phase diagrams for tight-binding models,}
J. Stat. Phys. {\bf 96}, 545-611 (1999).

\bibitem{4}
 F. Dyson, E. H. Lieb and B. Simon,
{\em Phase transitions in quantum spin systems with isotropic and non-isotropic interactions,}
J. Stat. Phys.  {\bf 18}, 335 (1978).

\bibitem{3}
J.~Fr\"ohlich, B.~Simon and T.~Spencer,
{\em Infrared Bounds, Phase Transitions And Continuous Symmetry Breaking,}
Commun.\ Math.\ Phys.\  {\bf 50}, 79 (1976).


\bibitem{B}
N.D. Mermin, H. Wagner, {\em Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models},
Phys. Rev. Lett. {\bf 17}, 1133-36 (1966).


\bibitem{1}
W. Heisenberg, Z. Physik {\bf 49}, 619 (1928).

F. Bloch,
Z. Physik {\bf 61}, 206 (1930).

E.C. Stoner, Proc. R. Soc. {\bf A165}, 372 (1938).

F.J. Dyson,
{\em General Theory of Spin-Wave Interactions,}
Phys. Rev. {\bf 102}, 1217-1230 (1956).

F.J. Dyson,
{\em Thermodynamic Behavior of an Ideal Ferromagnet,}
Phys. Rev. {\bf 102}, 1230-1244 (1956).

L.D. Landau, E.M. Lifshitz,
{\em Course of Theoretical Physics}
Volume 9, {\em Statistical Physics}, Part 2, by E.M. Lifshitz and L.P. Pitajewski, London: Pergamon 1981.

\bibitem{2}
E.~Brezin, J.~C.~Le Guillou and J.~Zinn-Justin,
{\em Field Theoretical Approach To Critical Phenomena,}
in
{\em Phase Transitions and Critical Phenomena,}  C. Domb and M.S. Green (eds.) Volume 6, 125-247, London: Academic Press 1976.

D.J. Amit,
{\em Field Theory, the Renormalization Group and Critical Phenomena},
New York a.o.: McGraw-Hill 1978.


\bibitem{C}
A. Mielke, H. Tasaki, {\em Ferromagnetism in the Hubbard model},
Commun. Math. Phys. {\bf 158}, 341-71 (1993).

\bibitem{D}
H. Tasaki, {\em Ferromagnetism in the Hubbard model: A constructive approach},
Commun. Math. Phys. {\bf 242}, 445-72  (2003).

\bibitem{E}
J. Fr\"ohlich, D. Ueltschi, {\em Hund's Rule and Metallic Ferromagnetism},
J. Stat. Phys.  {\bf 118}, 973-996 (2005).


\bibitem{7}
W.~Weber, S.~Riesen and H.~C.~Siegmann,
{\em Magnetization Precession by Hot Spin Injection},
Science {\bf 291}, 1015, (2001).

\bibitem{8}
D. Oberli,
{\em The ferromagnetic spin filter,}
PhD thesis ETH Z\"urich, Nr. 12933, (1998).
% http://e-collection.ethbib.ethz.ch/show?type=diss&nr=12933.

\bibitem{9}
J.~Fr\"ohlich, U.~M.~Studer and E.~Thiran,
{\em Quantum theory of large systems of nonrelativistic matter}, in
%Les Houches Summer School, Session 62: Fluctuating Geometries in Statistical Mechanics and Field Theory,
{\em Fluctuating geometries in statistical mechanics and field theory} (Les Houches 1994), F. David, P. Ginsparg, J. Zinn-Justin (eds.); Elsevier 1996,
[arXiv:cond-mat/9508062].

\bibitem{13}
E. Fradkin,
{\em Field theories of condensed matter systems},
Redwood City, California a.o.: Addison-Wesley 1994.

\bibitem{F}
J. Reuteler, {\em QM particles scattering off a dynamical target, and decoherence}, ETH-diploma thesis, spring 2005.

\bibitem{11}
T.~Chen, J.~Fr\"ohlich and M.~Seifert,
{\em Renormalization group methods: Landau-Fermi liquid and BCS superconductor,} in
{\em Fluctuating geometries in statistical mechanics and field theory}, F. David, P. Ginsparg, J. Zinn-Justin (eds.); Elsevier 1996,
[arXiv:cond-mat/9508063].

\bibitem{G}
M. Azam, L. Ferrari, J. Fr\"ohlich, to appear.

\bibitem{17}
H.~Leutwyler,
{\em Nonrelativistic effective Lagrangians,}
Phys.\ Rev.\ D {\bf 49}  3033 (1994),
[arXiv:hep-ph/9311264].


\bibitem{14}
J. Fr\"ohlich, R. Israel, E.H. Lieb, B. Simon,
{\em Phase Transitions and Reflection Positivity I. General Theory and Long Range Lattice Models,}
Commun. Math. Phys. {\bf 62}, 1-34, (1978).

\bibitem{14II}
J. Fr\"ohlich,
{\em Mathematical Physics of Phase Transitions and Symmetry Breaking},
Bulletin of the American Mathematical Society, {\bf 84}, 165-193 (1978).

\bibitem{10}
D. Ruelle,
{\em Statistical mechanics: rigorous results,}
Singapore: World Scientific, 1999.



\bibitem{12}
L. Birke, J. Fr\"ohlich,
{\em KMS, ETC.}
Rev. Math. Phys. {\bf 14}, 829-872, (2002),
[arXiv: math-ph/0204023].



\bibitem{15}
J.~Fr\"ohlich and T.~Spencer,
{\em Phase transitions in statistical mechanics and quantum field theory,}
in {\em New Developments in Quantum Field Theory and Statistical Mechanics}; M. L\'evy, P. Mitter (eds.), New York, London: Plenum 1977.

\bibitem{16}
E.H. Lieb,
{\em The Classical Limit of Quantum Spin Systems},
Commun. Math. Phys. {\bf 31}, 327-340, (1973).

\bibitem{18}
J.~Derezinski, V.~Jaksic and C.~Pillet,
{\em Perturbation theory of W*-dynamics, Liouvilleans and KMS-states,}
Rev.\ Math.\ Phys.\  {\bf 15} 447, (2003).

V.~Bach, J.~Fr\"ohlich and I.~M.~Sigal,
{\em Return to equilibrium,}
J.\ Math.\ Phys.\  {\bf 41} 3985 (2000) .

J. Fr\"ohlich, M. Merkli,
{\em Another Return of 'Return to Equilibrium'},
Commun. Math. Phys. {\bf 251}, 235-262, (2004).

\bibitem{Bach}
V. Bach, E. H. Lieb, J.P. Solovej, {\em Generalized Hartree-Fock Theory and the Hubbard Model},
[arXiv:cond-mat/9312044].

\end{thebibliography}