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\begin{document}
\title{Gauge symmetries\\ in the Heisenberg model of a magnet}
\author{T. Lulek and W. Florek\\
	 Division of Mathematical Physics\\
	 The Institute of Physics\\
	 A.Mickiewicz University\\
	 Matejki 48/49\\
	 60-769 Pozna\'n, Poland}


\maketitle
\begin{abstract}
A relation between geometric and gauge symmetries for the Heisenberg model of
magnet is pointed out. The space of quantum states of the magnet exhibits the
structure of a bundle, with the base consisting of nodes of the crystal, and
the typical fiber spanned on the set of the single-node spin projections. The
geometric symmetry group acts on the base, the gauge group --- on the typical
fiber, and all combined operations form the wreath product. In particular,
all global gauge transformations yield the direct product group --- a
subgroup of the wreath product.
\end{abstract}

\section{Introduction}
The Heisenberg model of a magnet exhibits a rich combinatorial structure,
associated with a group enumeration of various classes of magnetic
configurations \cite{ref1,ref2}. These classes are orbits of certain groups
of the magnet. The most obvious symmetry is the geometric one, consisting of
such operations that preserve the distances between magnetic nodes. Here we
aim to point out some other operations, called gauge symmetries \cite{ref3},
which involve transformations of internal spaces of each magnetic node. They
are well described in terms of fiber structure of space of all quantum states
of the magnet \cite{ref4,ref5}. We also discuss here the relation of gauge
symmetries to wreath products.

\section{The Heisenberg model and combinatorics}
Kinematics of the Heisenberg model of a magnetic crystal is specified by the
space of all its quantum states. Let $X$ be the set of all nodes of the
crystal, and let each node carries a spin $s$. Let $Y$ be the set of all
projections of the single-node spin. Let
\begin{equation}
|X| = n,\;\; |Y|=m,
\end{equation}
so that $n$ is the number of the crystal nodes, and $m=2s+1$ defines the
value of the spin $s$. Then the set
\begin{equation}
Y^X=:\{f:X\rightarrow Y\}
\end{equation}
of all mappings
\begin{equation}
f=\left|i_1,\ldots,i_n\right>,\;\;\;\;i_j\in Y,\;\;j\in X,
\label{eq3}
\end{equation}
is the set of all magnetic configurations. Eq.\ (\ref{eq3}) displays the
Dirac ket notation for a magnetic configuration $f$. The linear closure
\begin{equation}
L = \mbox{\rm lc}_{\mbox{\scriptsize \mathc}}Y^X 
\end{equation}
of the set $Y^X$ over the field \mathc\ of complex numbers is a linear space.
This space is equipped with the unitary structure by demand that the set
$Y^X$ is an orthonormal basis in it. $L$ is the space of all quantum states
of the magnet.

Combinatorial enumeration of orbits is given in terms of various group
actions. Let $S_n$ and $S_m$ be the symmetric group on the set $X$ and $Y$,
respectively. Then the defining actions $U:S_n\times X\rightarrow X$ and
$V:S_m\times Y\rightarrow Y$ are naturally lifted to $P:S_n\times
Y^X\rightarrow Y^X$ and $Q:S_m\times Y^X\rightarrow Y^X$. Subductions of
actions $P$ and/or $Q$ to various subgroups of $S_n$ and/or $S_m$ yield
orbits, associated with appropriate symmetries.

\section{Geometric symmetries}
Let $H\subset S_n$ be the geometric symmetry group of the crystal (it can be
a point group or a finite extension of translation group, resulting from the
periodic Born--von K\'arm\'an boundary conditions). Similarly, let 
$G\subset S_m$ be a subgroup of the single-node group $S_m$, e.g.\ the
two-element group involving the time reversal. In such a case, the direct
product $H\times G$ describes the total space-time symmetry of the magnet.
This symmetry is realized in terms of the subduced action
$(P\downarrow H)\times(Q\downarrow G)$.

The geometric, and --- more generally --- space-time symmetry of the magnet
is therefore related to the direct product group $H\times G$. A more general
combinatorial construction is the wreath product $G\wr H$. There is thus a
temptation to ask whether the action of the wreath product $G\wr H$ on the
set $Y^X$ of all magnetic configurations has also a physical meaning? We
propose an answer in Sec.\ \ref{wrprod}.

\section{Fiber structure of the space $L$ of quantum states of the magnet}
Let 
\begin{equation}
W = \mbox{\rm lc}_{\mbox{\scriptsize \mathc}}Y 
\end{equation}
be the linear unitary space of quantum states of single-node spin $s$, with
$Y$ as its orthonormal basis. We can think of $X$ as the base of a bundle $E$
with the typical fiber $W$. To be more specific, let 
$\phi_j:W\rightarrow W_j$, $j\in X$, be a copying isomorphism, which produces a
faithful copy of the space $W$, centered at the node $j$ of the crystal $X$.
Then the collection of all such copies,
\begin{equation}
E=\bigcup_{j\in X}W_j
\end{equation}
is the fiber bundle with the base $X$ and the typical fiber $W$. The
canonical bundle projection $p:E\rightarrow X$ is given by
\begin{equation}
p(e)=j\hspace{2ex}\mbox{\rm for}\hspace{2ex}e\in W_j\subset E.
\end{equation}
Every mapping $\psi:X\rightarrow E$ with the property
\begin{equation}
p\circ \psi = \mbox{\rm id}_X,
\end{equation}
where $\mbox{\rm id}_X$ is the identity mapping on the base $X$, is called a
section of the bundle $E$. We observe that if
\begin{equation}
\psi(j)=i_j,\;\;j\in X,
\end{equation}
then $\psi\in Y^X$. Thus some sections of the bundle $E$ can be identified as
magnetic configurations.

Clearly, the action $V:S_m\times Y\rightarrow Y$ can be extended to the
action $V':U(m)\times W\rightarrow W$  of the unitary group $U(m)$ on the
typical fiber $W$. The unitary group $U(m)$ is the maximal quantum symmetry
group for the single spin $s$.

\section{Gauge symmetries}
Now we are in a position to consider a subgroup $\Gamma\subset U(m)$ as the
gauge group. Some prominent examples from physics are $\Gamma =U(1)$ for
quantum electrodynamics, $\Gamma=SU(2)$ for isospin symmetry, and
$\Gamma=SU(3)$ for quarks.

We consider the action of the gauge group $\Gamma$ on the space $L$ of the
Heisenberg magnet. The simplest case is when each $g\in\Gamma$ acts in the
same way on each fiber $W_j$, i.e.\ when
\begin{equation}
(g\psi)(j)=\phi_j(g\phi_j^{-1}(\psi(j))),\;\;j\in X.
\label{gpsi}
\end{equation}
Eq.\ (\ref{gpsi}) defines a global gauge transformation in the space $L$,
imposed by the element $g\in\Gamma$. Clearly, combination of global gauge
transformations with geometric symmetries $H$ yields the action of the direct
product group $\Gamma\times H$ in the space $L$.

We proceed to introduce local gauge transformations. To this aim, we consider
a mapping $c:X\rightarrow\Gamma$, called a cochain, such that each element
$c(j)\in\Gamma$, $j\in X$, of the gauge group $\Gamma$ acts on its own fiber
$W_j$. Under such an action, each section $\psi\in L$ transforms to a new
section 
\begin{equation}
c\psi=\psi'\in L,
\end{equation}
given by
\begin{equation}
(c\psi)(j)=\phi_j(c(j)\phi_j^{-1}(\psi(j))),\;\;j\in X.
\label{cpsi}
\end{equation}
Eq.\ (\ref{cpsi}) defines the local gauge transformation in the space $L$. It
is worth to notice that $c\psi$ is not a composition of mappings, but only a
short notation for a new section $\psi'\in L$, the image of $\psi$ under the
local gauge transformation $c$.

The group of all local gauge transformations in the space $L$, imposed by the
gauge group $\Gamma$ coincides with the set
\begin{equation}
C^1(X,\Gamma)=\Gamma^X=\{c:X\rightarrow\Gamma\}
\label{c1}
\end{equation}
of all cochains on the set $X$, valued in the group $\Gamma$, with the group
multiplication imposed by $\Gamma$.


\section{Wreath product}
\label{wrprod}
Assume that the crystal $X$ is an orbit of the geometric symmetry group
$H\subset S_n$. Then the group (\ref{c1}) of all local gauge transformations
imposed by $\Gamma\subset U(m)$ can be identified as the base group of the
wreath product
\begin{equation}
\Gamma\wr H = \Gamma^X\Box H
\label{wrgh}
\end{equation}
of the gauge symmetry group $\Gamma$ by the permutation group $H$ on the set
$X$ of nodes of the crystal. We have thus arrived at a clear physical
interpretation of the wreath product (\ref{wrgh}). In particular, operations
$(c,1_H)\in\Gamma\wr H$ of the base group $\Gamma^X$ of the wreath product are
local gauge transformations. Operations of the form 
$(1_\Gamma,h)\in\Gamma\wr H$ coincide with purely geometric transformations,
whereas general operations $(c,h)\in\Gamma\wr H$ are combinations of
geometric and gauge symmetries.

\section{Conclusions}
We have shown a relation between the wreath product structure and gauge
symmetries of the Heisenberg model of a magnet. This relation is readily
expressed in terms of fiber bundle structure of the space $L$ of all quantum
states of the magnet. The geometric symmetries, described by a subgroup $H$
of the symmetric group $S_n$, are related to the base $X$ of the bundle $E$,
whereas the gauge group $\Gamma$ is the subgroup of the unitary group $U(m)$.
The former describes the spatial distribution of nodes of the crystal, whereas
the latter emerges from the internal symmetry of the typical fiber $W$,
spanned on the set $Y$ of single-node spin projections. All combinations of
geometric and gauge symmetries are given by the wreath product $\Gamma\wr H$.

\begin{thebibliography}{9}
\bibitem{ref1} T. Lulek, {\it Group enumeration, fiber bundles and a finite
  Heisenberg magnet}, Acta Magnetica {\bf VI} (1989) 29.

\bibitem{ref2} A. Kerber, B. Lulek, T. Lulek, {\it Group actions,
  configurations and finite states}, in: W. Florek, T. Lulek, and M. Mucha
  (Eds.), {\it Symmetry and Structural Properties of Condensed Matter}, World
  Sci., Singapore 1991, p. 3.

\bibitem{ref3} N.P. Konopleva (Ed.), {\it Quantum Theory of Gauge Fields},
  Mir, Moscow 1977 (in Russian).

\bibitem{ref4} W. Florek and T. Lulek, {\it Symmetry properties of the
  density of states in the Brillouin zone for a one-dimensional periodic
  Heisenberg magnet}, J. Phys. A (Math. Gen.) {\bf 20} (1987) 1921.

\bibitem{ref5} B. Lulek, {\it Density of states in the Brillouin zone for a
  finite magnetic linear chain with a single impurity}, Acta Phys. Pol. {\bf
  A74} (1988) 453.

\bibitem{ref6} A. Kerber, {\it Representations of Permutation Groups I, II},
  Lect. Notes in Math., vols. 240 and 495, Springer-Verlag, Berlin 1971,
  1974.

\bibitem{ref7} G. James and A. Kerber, {\it The Representation Theory of the
  Symmetric Group}, Adison--Wesley, Reading 1981.
\end{thebibliography}


\end{document}
