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PACS numbers: 75.40.Gb, 67.57.Lm, 05.70.Ln
Email: spitzer@math.ku.dk
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Landau-Lifshitz equations, macroscopic mean-field limit
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\documentstyle[prl,aps,twocolumn,amssymb]{revtex}
%%%%%%%%%%%%%%%%%%%%%%%%%%%% newcommands %%%%%%%%%%%%%%%%%%%%%%%%%
\def\a{\alpha}
\def\b{\beta}
\def\c{\gamma}
\def\d{\delta}
\def\e{\epsilon}
\def\s{\sigma}
\def\w{\omega}
\def\D{\Delta}
\def\W{\Omega}
\def\l{\lambda}
\def\p{\partial}
\def\fv{\frak v}
\def\fF{\mathfrak F}
\def\oe{\omega^\epsilon}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\beax}{\begin{eqnarray*}}
\newcommand{\eeax}{\end{eqnarray*}}
\newcommand{\op}[1]{\mbox{\sf #1}}
\newcommand{\komm}[2]{\left[#1,#2\right]}
\begin{document}
\draft
\twocolumn[\hsize\textwidth\columnwidth\hsize\csname @twocolumnfalse\endcsname
\title{Time Evolution of Spin Waves}
\author{Michael Moser}
\address{Justgasse 29/12/3/12, A-1210 Vienna, Austria}
\author{Alexander Prets}
\address{Institut f\"ur Theoretische Physik, Universit\"at Wien,
Boltzmanngasse 5, A-1090 Vienna, Austria}
\author{Wolfgang L Spitzer}
\address{Department of Mathematics, University of Copenhagen, Universitetsparken 5,
DK-2100 Copenhagen, Denmark}
%
\date{June, 1999}
\maketitle
\begin{abstract} A rigorous derivation of macroscopic spin-wave equations
is demonstrated. We introduce a macroscopic mean-field limit and derive the
so-called Landau-Lifshitz equations for spin waves. We first discuss the
ferromagnetic Heisenberg model at $T=0$ and finally extend our analysis to
general spin hamiltonians for the same class of ferromagnetic ground
states.
\end{abstract}
\pacs{PACS numbers: 75.40.Gb, 67.57.Lm, 05.70.Ln}
]
\narrowtext
\noindent An important problem in statistical physics is to explain the macroscopic
behavior of matter from first principles, in particular equations of motion of
macroscopic observables. There is by now an extensive literature on classical
systems and we want to refer to a standard treatment by Spohn \cite{spohn91}.
The general idea is that macroscopic equations emerge from microscopic equations
of motion (say Newton's or Schr\"odinger's) for the constituents of matter by
performing some kind of scaling limit. There are different types of
scaling limits whose applicability is to be justified by the physical
circumstances.
In this paper we study equations of motion for the magnetization in certain
magnetic samples. Long ago Landau and Lifshitz \cite{landau35} derived such
equations. In case of ferromagnets it says that the change of the magnetization
vector, $\vec{M}(r,t)$\footnote[1]{$\vec{M}:(r,t) \in {\mathbb R}^d
\times {\Bbb R}\mapsto \vec{M}(r,t)\in {\mathbb R}^3$. We set Bohr's magneton
$\mu_0=\hslash/2$, which puts the constant on the right
hand side equal to 1.}, is simply given by rotation, i.e.
\be
\frac{\partial\vec{M}({r},t)}{\partial t} =
\vec{M}({r},t)\,\times\,\vec{H}_{\rm eff}({r},t).
\label{l1}
\ee
Here, the angular velocity is equal to an effective magnetic field,
$\vec{H}_{\rm eff}({r},t)$, which originates from the microscopic
interactions of the spins. If they, say,
interact only via ferromagnetic exchange interaction \cite{akhiezer68} with
coupling function $J$, than $\vec{H}_{\rm eff}({r},t)$ is given by
\be
\vec{H}_{\rm eff}({r},t) = \int dr'\,J({r},{r}')\,
\vec{M}(r',t).
\label{l2}
\ee
Expanding $\vec{M}(r',t)$ around $(r,t)$ and considering an isotropic potential,
$J(r,r')=J(|r-r'|)$, $\vec{H}_{\rm eff}$ becomes in lowest
order proportional to the laplacian $\D\vec{M}({r},t)$, and (\ref{l1})
simplifies to
\be\label{l3}
\frac{\partial}{\partial t}\vec{M}(r,t)=\a\,\vec{M}(r,t) \,\times\, \Delta
\vec{M}(r,t)
\ee
with
$$
\a=\frac{1}{6}\int dr\,J(|{r}|)\,r^2.
$$
Usually, (3) is called the Landau-Lifshitz equation.
We are aware of only two other sources \cite{akhiezer68,kittel}, where
a derivation of equation (\ref{l3}) is demonstrated but, nevertheless, is partly
based on non-rigorous arguments; some of them (e.g. magnon picture),
however, are extremely important, influential and still a major challenge in
this field.
We will derive the more general equation (\ref{l1}) by applying what we call
a macroscopic mean-field limit \cite{spohn91,demasi91,spohn94}. In this context
we mention that the Vlasov equation for a quantum mean-field
system has been proven by Narnhofer and Sewell \cite{narnhofer}.
One might think that a direct derivation of equation (\ref{l3}) is
easier than of (\ref{l1}), but the former does not seem to originate from a
microscopic model \cite{prets98}. This can be compared with a recent work on
the derivation of the Cahn-Hilliard equation \cite{giacomin}. We want to
emphasize that we did not start with a mean-field model in the first place but
we were forced, both for physical and mathematical reasons,
to do so. We hope that this will become clear in our presentation.
Admittedly, other approaches may directly lead to equation (\ref{l3}).
Now we provide the set-up for the derivation of equation (\ref{l1}).
Since magnetism is a purely quantum mechanical effect of interacting
spin systems we clearly have to formulate a quantum mechanical model
\cite{spitzer96}. For mathematical convenience we consider a bounded sample,
say the cube $[0,1]^d$. In order to describe the microscopic situation we
enlarge the cube by a factor $\e^{-1},\e>0$.
Finally, we let $\e\downarrow0$. $K(\e) = [0,\e^{-1}]^d \cap {\mathbb Z}^d$
represents the ferromagnet from a microscopic point of view.
We use the general ferromagnetic Heisenberg-Kac hamiltonian, which is motivated
by the mean-field structure of (\ref{l2}).
\be
H^\e=-{\frac{1}{2}}\sum_{x,y\in K(\e)} J_{xy}^\e \, S_x^\a\,S_y^\a,
\label{l4}
\ee
to describe the microscopic exchange interaction energy between spins in
$K(\e)$. As usual, $H^\e$ acts on the $|K(\e)|$-fold tensor product of the
space ${\mathbb C}^{2S+1}$. $S$ is some fixed (but arbitrary) half-integer.
Spin operators $\vec{S_x}=(S_x^1,S_x^2,S_x^3)$ at
lattice site $x$ (acting on ${\mathbb C}^{2S+1}$) satisfy
the usual commutation relations
$$
\komm{S_x^\a}{S_y^\b}=i\,\d_{xy}\,\e_{\a\b\c}\,S_y^\c.
$$
We scale the interaction potential like $J^\e_{xy} = {\e}^{+d}J(\e x, \e y)$,
which can be viewed as a Lebowitz-Penrose approximation \cite{LP} to the true
interaction; the factor $\e^d$ in front of the sum provides us with an extensive
energy function (see also another discussion below). Here, the function $J$ on
$[0,1]^{2d}$ is assumed to be non-negative but not necessarily symmetric.
The general idea relating the different levels of description is to define
the {\it macroscopic} magnetization pro\-file $M^\c(r,t)$ as the $\e$-scaling
limit of the expectation value of the time dependent {\it microscopic} spin
operators $S^\c_x(t)$, see (\ref{lim}).
Thus the second main ingredient in our analysis, besides the hamiltonian
(\ref{l4}), is to specify a class of (initial) states, denoted hereafter by
$\fF$. It is clear that (\ref{l1}) can only be correct at low temperature.
As is well-known spins are parallel to each other in the (highest weight)
ground state of the (isotropic) ferromagnetic Heisenberg model.
In our states spins will remain almost parallel over short (compared to
$\e^{-1}$) distances but vary over distances of
order $\e^{-1}$, i.e. we locally describe $T=0$ equilibrium states as
\be \label{l8}
{\fF}=\Big\{(\oe)_{\e>0}: \oe= \bigotimes_{x\in K(\e)} \oe_x :
\oe_x(S_x^\c) = M^\c(\e x)\Big\}\nonumber.
\ee
$\oe_x$ is a state on the matrix algebra over ${\mathbb C}^{2S+1}$; in the case
$S=1/2$ the states $\oe_x$ are uniquely determined by $M^\c$ itself.
We rescale space by $\e$ but we {\em do {not} rescale time.} This particular
space vs time scaling limit fits nicely with Landau's idea of classical spin
waves of large wave-length, since the microscopic spin-wave length goes as
${\cal O}(\e^{-1})$.
In what follows we require
$M^\c(r,0),\phi(r)$ and $J(r,r')$ to be continuous functions on
the cube $[0,1]^d$. We now show that at least for
$0<|t|1
\nonumber
\eea
with
$$
T^{\a}_m=\sum_{k=1}^m\underbrace{\op{1}\otimes\dots
\otimes\op{1}}_{k-1\,\mbox{{\scriptsize times}}}\otimes\left(\e^{\a}
\,J_{y_ky_{m+1}}\right)\otimes\underbrace{\op{1}\otimes\dots\otimes
\op{1}}_{m-k\,\mbox{{\scriptsize times}}}.
\label{l9}
$$
%$\d_{\c\a}$ is the usual $\delta$-function, and
$\e^\a$ denotes the matrix of the
$\e$-tensor, $\e_{123}=1$. The rest term, $\hat{R}_n^\e$, is given by
\bea\label{l5c}
\hat{R}^\e_n&=&\sum_{k=0}^nK^{n-k}(H^\e)\,R_k^\e,
\\
R_k^\e&=&
\frac{1}{4}(-i)^k{\sum_{y_2,\dots,y_k\in K(\e)}}^
{\hspace*{-1.5em}'}\hspace{1.2em}\sum_{m=1}^k P_k(\vec{y}_k,
y_m)^\c_{\vec{\a}_k,\a_m}\hat{S}_{y_m}^{\a_m},\nonumber
\eea
with $\hat{S}_{y_m}^{\a_m}=\prod_{m\neq l=1}^kS_{y_l}^{\a_l}$ and
$R_0^\e=R_1^\e=0$. Formulas (\ref{l5b}-\ref{l5c}) can be proven by induction.
Since there are more $\e^d$-factors than sums the operator norm
of the rest term vanishes as $\e\downarrow0$,
\be
\lim_{\e\downarrow0}||\hat{R}_n^\e||=0.
\label{l5d}
\ee
We first show that (for $\c=1,2,3$) the maps
\[\phi\mapsto L^\c(\phi,t):=\lim_{\e\downarrow0}\e^{d}\sum_{x\in K(\e)}
\phi(\e x)\,\oe(S^\c_x(t))
\]
are continuous linear functionals on the Banach space $C([0,1]^d,\|\cdot
\|_\infty)$ (of continuous functions equiped with the maximum norm) for
sufficiently small $|t|$ (depending on the interaction function $J$ and on the
spin $S$), which guarantees the existence of measures $dM^\c(r,t)$. This is
accomplished by roughly estimating $\e^{d}\sum_{x\in K(\e)}|\phi(\e x)\,
\oe(S^\c_x(t))|$ uniformly in $\e$ by $\|\phi\|_1 \cdot \|M\|_\infty (1-2t\,
\|J\|_\infty\,\|M\|_\infty)^{-1}$ (up to ${\cal O}(\e)$) which holds at least
for $|t|^{-1}>\|J\|_\infty\,\|M\|_\infty=:C^{-1}$; note that there are $2^nn!$
contributing terms. This also shows analyticity for $|t|