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\title{A New Representation For (1+1)-Dimensional
Landau-Lifshitz Model}
\author{ S.B. Rutkevich}
\date{}
\begin{document}
\begin{titlepage}
\maketitle
\begin{center}
{\footnotesize Institute of Physics of Solids and
Semiconductors,\\
P. Brovki St. 17, 220072 Minsk, Belarus\\
{\it E-mail: lttt@ifttp.basnet.minsk.by}}
\vfil
ABSTRACT
\end{center}
\begin{quote}
Recently an exactly solvable classical Hamiltonian model was
obtained, which appears in analysis of the correlation function in a
free-fermion model. It is proved here, that the above mentioned
Hamiltonian model is equivalent to the Landau-Lifshitz model of
the one-dimensional biaxial ferromagnet.
\end{quote}
\end{titlepage}
%%
It is widely excepted that correlation functions in quantum
integrable models are closely related with some classical
Hamiltonian models, which in turn can be solved exactly by the
inverse scattering method (see \cite{J, Kor, Ess}). Recently such
a relationship was established for a certain free-fermion model
\cite{rut}, which is equivalent to the XY-spin chain model in
the double scaling limit
\footnote{ The double scaling limit of the
XY-model was introduced by Vaidya and Tracy \cite{V},
and studied in detail by Jimbo {\it et al} \cite{J}.}. It was
shown that the two-point correlation function in the free-fermion
model is
related with the classical integrable model defined by the
Hamiltonian \cite{rut}:
\begin{equation}
\label{E}
{\cal H}={1\over 2} \int {\rm d}x
\biggl\{ \pi^2 (x)[u'^{\, 2} (y)+(1/4) R(u)] +
\frac {[u''(x)+(1/8)\,d R(u)/d u]^2} {[u'^{\, 2}(x)+(1/4) R(u)]}
- k^2 \, u^2(x) \biggl\}
\end{equation}
where
\begin{equation}
\label{R}
R(u)=(1-u^2)(1-k^2 u^2),
\end{equation}
$u(x)$ and $\pi(x)$ are the canonical coordinate and
momentum functions, and $k$ is
the elliptic module.
A natural question arises in connection with Hamiltonian (\ref{E}):
whether it describes a new exactly solvable model, or it
corresponds to some well-known model, written in unusual
variables,
however. The answer on this question is obtained in the present
letter. We prove that above classical model is canonically
equivalent to the complex generalization of the Landau-Lifshitz
model of the biaxial one-dimensional ferromagnet.
Landau-Lifshitz model is defined by the Hamiltonian density
\cite{Fad}
\begin{equation}
\label{HL}
H_{LL}(x) ={1\over 2}\Biggl[\biggl(\frac{\partial
{\bf S} }{\partial x}\biggl)^2-
J({\bf S})\Biggl]
\end{equation}
where ${\bf S}(x)=\biggl(S_1 (x), S_2 (x), S_3 (x)
\biggl)$ is a real vector on a unit sphere
${\bf S}(x)\in${\sf S}$^2\subset${\bf R}$^3$:
\begin{equation}
\label{S}
{\bf S}^2 (x)=\sum_{a=1}^3 S_a^2 (x) =1,
\end{equation}
$J({\bf S})$ is the diagonal quadratic form
\begin{eqnarray}
\label{J}
J({\bf S})=J_1 S_1^2+J_2 S_2^2+J_3 S_3^2
\nonumber \\
J_1\le J_2\le J_3,\qquad J_1+J_2+J_3=0
\end{eqnarray}
The equation of motion is Hamiltonian
\begin{equation}
\label{em}
\dot {\bf S}=\{{\cal H}_{LL},{\bf S}\}
\end{equation}
where ${\cal H}_{LL}$ is the Hamiltonian:
${\cal H}_{LL} = \int {\rm d}x \, H_{LL} (x)$.
The Poisson structure is induced by the Poissin brackets
\begin{equation}
\label{P}
\{ S_a (x), S_b (y)\} = -\epsilon_{abc} S_c (x) \delta(x-y)
\end{equation}
Above model is known as a "uiversal" (in a certain sense) integrable
model, which contains sine-Gordon and nonlinear Schr\"odinger
models as limit cases. We shall consider the comlex generalization
of the model (\ref{HL}-\ref{P}) supposing , that ${\bf S}(x)$
is a vector on a {\it complex} unit sphere (\ref{S}): {\sf S}$^2
\subset$
{\bf C}$^3$ .
First, let us introduce variables $\psi (x)$ and $\psi^* (x)$ on
the sphere by the relations
\[
\psi=\frac{S_1 +i S_2}{(1-S_3)^{1/2}},\quad
\psi^*=\frac{S_1 -i S_2}{(1-S_3)^{1/2}}
\]
Since the Poisson bracket on these functions is canonical
\[
\{\psi(x),\psi^* (y) \}=-i \delta (x-y)
\]
apart from the factor $-i$, the action functional ${\cal S}$ for the
model (\ref{HL}-\ref{P}) can be written as
\[
{\cal S} = \int_{t_1}^{t_2} {\rm d}t \int {\rm d} x L(x,t)
\]
with the Lagrangian density $ L(x,t)$ given by
\[
L= -i \psi^* \dot \psi - H_{LL}
\]
Next, rewrite this Lagrangian density in terms of another pair of
variables $w(x)$ and $w^*(x)$ which give the stereographic
projection of the sphere {\sf S}$^2$:
\begin{equation}
\label{st}
S_1 + i S_2 = \frac {2w}{1+w w^*}, \quad
S_1 - i S_2 = \frac {2w^*}{1+w w^*}, \quad
S_3=\frac{1-ww^*}{1+ww^*}
\end{equation}
or, equivalently
\begin{equation}
\psi =\biggl[\frac{2w}{w^*(1+ww^*)}\biggl]^{1/2}, \quad
\psi^* =\biggl[\frac{2w^*}{w(1+ww^*)}\biggl]^{1/2}
\end{equation}
The resulting expression takes the form
\begin{equation}
L \simeq - \frac{2i\dot w}{w(1+ww^*)}-
2\frac{(w'/w)'}{(1+ww^*)}
+2\frac{w^*(w')^2}{w(1+ww^*)^2}-
{1\over 2}\frac{(J_2-J_1)(w^2+w^{*2})+6J_3
ww^*}{(1+ww^*)^2}
\label{ld}
\end{equation}
>From now on the equivalence $\simeq$ implies that we ignore
terms having
the structure of full derivatives in time or in coordinate.
It shoulg be pointed out, that variables $w$ and $w^*$ which were
complex conjugate to each other when ${\bf S}(x)\in${\bf
R}$^3$, become independent ones in the considered complex case
${\bf S}(x)\in${\bf C}$^3$. Varying the action corresponding to
the Lagrangian density
(\ref{ld}) with respect to $w^*$, one obtains
\begin{equation}
\label{emm}
(-i\dot w - w'')(1+ww^*)+2w^*(w')^2-\frac{J_2-J_1}{2} (w^3-
w^*)+{3\over 2} J_3 w(1-ww^*)=0
\end{equation}
This representation of the Landau-Lifshitz equation has been
widely used in study of its soliton and multisoliton solutions (see
\cite{Kos} and references therein).
We would like to note the following. Equality (\ref{emm}) is a
complicated nonlinear partial differential equation with respect to
the function $w(x)$. However, it becomes an elementary linear
algebraic equation when regarded with respect to the function
$w^*(x)$. Eliminating $w^*(x)$ from the Lagrangian density
(\ref{ld}) by use of (\ref{emm}), we find after some algebra
\begin{equation}
\label{dd}
L \simeq -{1\over 2}\, \frac {\dot u^2 -[u''(x)+(1/2)\,
(d P(u)/d u)]^2}
{[u'^{\, 2}(x)+ P(u)]}+(J_2-J_1) \, u^2
\end{equation}
Here we have denoted
\[
P(u)=\frac{J_2-J_1}{4}\,(1+u^4) -\frac{3 J_3}{2}\, u^2 , \quad
u=1/w
\]
Rescaling of variables
\[
u \rightarrow {\sqrt k}\,u, \quad x\rightarrow {\sqrt
\frac{k}{J_2-
J_1}}\, x, \quad
t\rightarrow \frac{k}{J_2- J_1} \,t, \quad L\rightarrow
-\frac{J_2-J_1}{k}\, L
\]
with $k$ obeying the relation $k^2- \frac{6 J_3}{J_2-J_1}
k+1=0$ tranforms (\ref{dd}) to
\begin{equation}
\label{ddd}
L \simeq {1\over 2} \biggl\{\frac {\dot u^2 -[u''(x)+(1/8)\,d R(u)/d
u]^2}
{[u'^{\, 2}(x)+ (1/4)R(u)]}+k^2 \, u^2\biggl\}
\end{equation}
where we have used notation (\ref{R}) for $R(u)$. This is exactly
the Lagrangian density corresponding to the model (\ref{E}).
Thus, we have proved the equivalence of the complex
generalization of the Landau-Lifshitz model with the Hamiltonian
model (\ref{E}) obtained in reference \cite{rut}.
In conclusion we present another form of the Lagrangian density
(\ref{ddd}):
\begin{equation}
\label{ds}
L\simeq {1\over 2} \biggl\{\frac {\dot z^2 -z''^{\, 2}}
{1+z'^{\, 2}}-(1+3 z'^{\, 2}) \wp (z) \biggl\}
\end{equation}
Variables $u$ and $z$ are related by the formula
$u={\rm sn}[(z-2K)/2]$. Here ${\rm sn}$ denotes elliptic sinus
with elliptic
module $k$. The
function $u(z)$ has primitive periods $8K$ and $4iK'$ in the
complex z-plane. Primitive periods of the elliptic Weierstrass
function $\wp (z)$ in (\ref{ds}) are reduced by half and equel
$4K$ and
$2iK'$.
It is surprising, that the Landau-Lifshitz model allows such a
compact representation (\ref{ds}).
I wish to thank A.S. Kovalev and I.M. Babich for helpful discussion.
\begin{thebibliography}{7}
\bibitem{J}
M. Jimbo, T. Miwa, Y. Mori and M. Sato, {\it Physica} {\bf 1D}
(1980) 80
\bibitem{Kor}
V.~E. Korepin, A.~G. Izergin, and N.~M. Bogoliubov, 1993
{\it Quantum Inverse Scattering Method,
Correlation Functions and
Algebraic Bethe Ansatz}
(Cambridge University Press)
\bibitem{Ess}
F.H.L. E\ss{}ler, H. Frahm, A.B. Its and V.E. Korepin,
{\it J. Phys. A: Math. Gen} {\bf 29} (1996) 5916
\bibitem{rut}
S.B. Rutkevich, {\it J. Phys. A: Math. Gen} {\bf 30} (1997) 3883
\bibitem{V}
H.G. Vaidya and C.A. Tracy, {\it Phys. Letts.} {\bf 68A} (1978)
378
\bibitem{Fad} L.D. Faddeev and L.A. Takhtajan, 1986 {\it
Hamiltonian
Method in the Theory of Solitons} (Berlin: Springer)
\bibitem{Kos}
A.M. Kosevich, B.A. Ivanov and A.S. Kovalev, {\it Phys. Rep.}
{\bf 194} (1990) 117
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\end{thebibliography}
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\end{document}