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\begin{document}
\title{Impurities in the Heisenberg magnet\\ and the general recipe of Weyl}
\author{B. Lulek\\
 Division of Mathematical Physics, The Institute of Physics\\
 A.~Mickiewicz University, Pozna\'n, Poland}
\date{}

\maketitle
\begin{abstract}
The general recipe of Weyl is applied to the Heisenberg model of a
one-dimensional magnetic chain with impurities. It is found that impurities
break the obvious translational symmetry, but preserve the hidden one.
Rarefied bands emerge from irregular orbits of action of the translation
group. 
\end{abstract}

\section{Introduction}
We examine here an application of the general recipe of Weyl \cite{ref1,ref2}
to the Heisenberg model of a magnet \cite{ref3,ref4}. It has been shown that
the hidden symmetry group of Weyl's recipe yields some symmetries of the
distribution of quantum states of a magnet within the Brillouin zone
\cite{ref3}, and that these symmetries can be described in terms of rarefied
bands \cite{ref3,ref4}. These results, obtained for a crystal with the
perfect translational symmetry, can be discussed again for crystals with
broken periodicity, caused by impurities. We have discussed already the case
of a single impurity \cite{ref5}, and found an apparently puzzling result
that breaking of translational symmetry increases, rather than decreases, the
symmetry of this distribution. Here we proceed to discuss a
more general case of some impurities in the light of the recipe of
Weyl. In particular, we aim to point out that breaking of translational
symmetry is equivalent to decreasing of the obvious symmetry group, which
does not automatically imply lowering of the hidden symmetry \cite{ref6}.

\section{The recipe of Weyl}
Let us assume that we have a system with some {\it obvious symmetry},
described by a group $G$. A good example is a system that constitutes a
regular orbit of this group, so elements of the system can be labelled by
$g\in G$. The recipe of Weyl states that then all intrinsic features of the
system are derivable from the group $\mbox{\rm Aut }G$, the group of all
automorphisms of $G$. This group, $\mbox{\rm Aut }G$, is called the {\it
hidden symmetry}\/ group of the system.

\section{The homogeneous magnet}
Let
\begin{equation}
X=\{j\mid j=1,2,\ldots,n\}
\end{equation}
be the set of all $n$ nodes of a crystal. Within the Heisenberg model of a
magnet, one assumes that each node $j\in X$ is occupied by a spin $s$. Let
\be
Y=\{i\mid i=1,2,\ldots,m\}
\ee
be the set of labels of projections $-s,-s+1,\ldots,+s$ of the spin $s$, so
that $m=2s+1$. Any mapping $f:X\rightarrow Y$ is called a magnetic
configuration. It can be displayed in a form 
\begin{equation}
f=\left|i_1,\ldots,i_n\right>,\;\;\;\;i_j\in Y,\;\;j\in X.
\end{equation}
The set of all magnetic configurations is
\begin{equation}
Y^X=\{f:X\rightarrow Y\}.
\end{equation}

We assume that $n$ nodes of our crystal are arranged into a one-dimensional
finite chain, or, more precisely, that the set $X$ constitutes a regular
orbit of the cyclic group
\be
G=C_n,
\ee
the translational symmetry group of the magnet. We take $C_n$ as the obvious
symmetry group of Weyl's recipe.

We denote by
\be
B=\{ k=0,\pm1,\pm2,\ldots,\left\{
   \begin{array}{lr}
     \pm(n/2-1), n/2\;\; & {\rm for~ } n {\rm ~even}\\
     \pm(n-1)/2          & {\rm for~ } n {\rm ~odd}
   \end{array}\right. \}  
\ee
the set of labels of all irreducible representations of the group $C_n$ over
the field \mathc\ of complex numbers. This set is referred to as {\it the
Brillouin zone}\/ in condensed matter physics.

We proceed to define the distribution of quantum states of the magnet over
the Brillouin zone $B$. Let $P:S_n\times Y^X\rightarrow Y^X$ be the action of
the symmetric group $S_n$ on the set $Y^X$ of all magnetic configurations.
This action is defined by the formula 
\be
P(\sigma)=\left(
      \begin{array}{c} 
      f\\ f\circ\sigma^{-1} 
      \end{array}
      \right), \;\;f\in Y^X,\;\; \sigma\in S_n.
\ee
Let, moreover,
\be
L=\mbox{\rm lc}_{\mbox{\scriptsize \mathc}}Y^X
\ee
be the linear closure over the field \mathc, spanned on the set $Y^X$, with the
unitary structure imposed by requirement that $Y^X$ is an orthonormal basis
in $L$.  We denote the action of the group $S_n$ in the linear unitary space
$L$ by the same letter $P$. Under the embedding $C_n\subset S_n$ this action
subduces to
\be
P\downarrow C_n\cong\sum_{k\in B}\rho(k)\Gamma_k,
\label{dec}
\ee
where $\Gamma_k$ is an irreducible representation of $C_n$. The decomposition
(\ref{dec}) defines the mapping $\rho:B\rightarrow\mathz$ of $B$ into the
ring \mathz\ of integers, called the {\it distribution of quantum states}\/
of the magnet {\it over the Brillouin zone} $B$. This distribution can be
evaluated as
\cite{ref4,ref7,ref8} 
\be
\rho(k)=\frac{1}{n}\sum_{\kappa\in K(n)}\;n^\kappa 
  \frac{\varphi(\bar{\kappa})}{\varphi(\bar{\kappa'})}\mu(\bar{\kappa'}),
\label{dis}  
\ee
where
\be
K(n)=\{\kappa\in X\mid {\rm lcd}(n,\kappa)=\kappa\}
\ee
is the lattice of all divisors of the integer $n$,
\be
\bar{\kappa}=n/\kappa
\ee
is the divisor complementary to $\kappa$ in the lattice $K(n)$,
$\varphi:K(n)\rightarrow \mathz$ and $\mu:K(n)\rightarrow \mathz$ is the
Euler and M\"obius function, respectively, and
\begin{eqnarray}
\kappa' &=& {\rm lcd}(\kappa\kappa_0,n),\\
\kappa_0 &=& {\rm lcd}(|k|,n),
\end{eqnarray}
with lcd denoting the largest common divisor.

The symmetry of the distribution $\rho$ is described by 
\be
{\rm Aut}\,C_n=\{r\in X\mid{\rm lcd}(r,n)=1\},
\ee
the hidden symmetry group of the recipe of Weyl \cite{ref3,ref4,ref9}. Let
$\Psi: {\rm Aut}\,C_n\times B\rightarrow B$ be the action of ${\rm Aut}\,C_n$
on the Brillouin zone $B$, defined by the formula
\be
\Psi(r)=\left(
           \begin{array}{c}
             k\\ rk \bmod n
           \end{array}
        \right),\;\; k\in B,\;\; {\rm lcd}(r,n)=1.
\ee
The action $\Psi$ yields orbits
\be
B^{(\kappa)}=\{k\in B\mid {\rm lcd}(k,\kappa)=\kappa\},\;\;\kappa\in K(n),
\label{bkappa}
\ee
called {\it generalized stars}. Thus the Brillouin zone decomposes into
disjoint generalized stars as
\be
B=\bigcup_{\kappa\in K(n)}B^{(\kappa)}.
\ee

Theory of condensed matter bases heavily on the notion of a band. A {\it full
band} corresponds to a such distribution of system states over the Brillouin
zone $B$, in which each quasimomentum $k\in B$ is associated with exactly
one quantum state. It is, in fact, a quantum analog of the first Newton law
of dynamics. A {\it rarefied band} correspond to a distribution, in which one
quantum state corresponds to a subset $B_0$ of the Brillouin zone $B$,
whereas the complement $B\setminus B_0$ carries no state.

The Heisenberg model of a magnet with a finite set $X$ of nodes yields
rarefied bands, carried by subsets \cite{ref4}
\be
B_0 = B|\kappa = \{\xi\kappa\in B\mid\xi\in \mathz\},\;\; \kappa\in K(n).
\label{rband}
\ee
The subset $B|\kappa$ is called a $\kappa$-tuply rarefied Brillouin zone. It
corresponds to the crystal with $\bar{\kappa}=n/\kappa$ nodes \cite{ref6}. In
particular, $\kappa=1$ corresponds to a full band.

We notify two important conclusions on the distribution $\rho$ given by Eq.\
(\ref{dis}): (i) it is {\it inhomogeneous} in the Brillouin zone $B$, so that
the magnet has to exhibit some rarefied bands (actually, these rarefied bands
(\ref{rband}) are labelled by $\kappa\in K(n)$), (ii) the distribution $\rho$
is constant on each generalized star $B^{(\kappa)}$ given by Eq.\
(\ref{bkappa}). It is an intrinsic feature of this distribution. 

These two conclusions are, in fact, partial results of Weyl's recipe. They
reflect the fact that the symmetry of the distribution $\rho$ is determined
by the hidden symmetry group Aut\,$C_n$.

\section{Impurities}
Now we assume that the subset $X_0\subset X$ is occupied by `host' atoms with
the spin $s$ ($m=2s+1$), whereas its complement $X'=X\setminus X_0$ is
occupied by `impurities' with the spin $s'\neq s$. Thus
\be
X=X_0\cup X',
\ee
and, accordingly,
\be
n=n_0+n'.
\ee
Thus the geometric distribution of nodes is the same as for the homogeneous
magnet, and the change consists in chemical occupation. The main difficulty
is caused by breaking of translational symmetry $C_n$ of the magnet by
impurities, so the notion of the Brillouin zone looses its original meaning.

Let the embedding $X'\subset X$ define the chemical configuration of the
magnet (i.e.\ the distribution of impurities). The set of all magnetic
configurations is now
\be
\Phi(X'\subset X)=Y_0^{X_0} \times Y'^{X'}
\ee
and depends on the chemical configuration of the magnet. Hence
\be
Y_0=Y,\;\;\; Y'=\{i\mid i=1,2,\ldots,m'\},\;\;m'=2s'+1.
\ee

The notion of quasimomentum and the Brillouin zone is restored by introducing
{\it the ensemble of magnets}, with the extended set of magnetic configurations
\be
\Phiens=\bigcup_{\sigma\in C_n}\;\Phi(\sigma(X')\subset X).
\ee
Obviously, the set $\Phi(X'\subset X)$ is not closed under the action of the
group $C_n$, whereas the set $\Phiens$ already is. The corresponding
action $\Pens:C_n\times\Phiens\rightarrow\Phiens$ can be
decomposed as 
\be
\Pens\cong\bigoplus_{k\in B} m(\Pens,k)\Gamma_k,
\ee
defining thus the average distribution $\rhoav:B \rightarrow
\mathq$ for the ensemble as
\be
\rhoav(k)=m(\Pens,k)/n,\;\; k\in B,
\ee
with \mathq\ being
the field of rational numbers.

For the case of single impurity, i.e.\ $n'=|X'|=1$, we obtain
\be
\rhoav=\frac{m^{n-1}m'}{n},\;\; k\in B.
\label{disav}
\ee
This distribution is thus homogeneous in the Brillouin zone. This result is
apparently puzzling, since the translational symmetry of the homogeneous
magnet yields an inhomogeneity of the distribution $\rho$ (cf.\ Eq.\
(\ref{dis})), whereas breaking of this symmetry by a single impurity implies
a homogeneous distribution $\rhoav$ (cf.\ Eq.\ (\ref{disav})). Thus it
looks that breaking of magnet symmetry increases the symmetry of quantum
state distribution in the Brillouin zone. Actually, this effect should be
attributed to the procedure of averaging over the ensemble, rather than
breaking of translational symmetry.

We conclude that average over the ensemble of magnets differing mutually by
the position of a single impurity (i) restores the quantum number $k$ of
quasimomentum and the Brillouin zone, (ii) wipes out the inhomogeneity of the
distribution $\rho$, (iii) preserves the hidden symmetry Aut\,$C_n$. In
particular, $\rhoav$ remains constant on each generalized star
$B^{(\kappa)}$, $\kappa\in K(n)$.

In general, the distribution $\rhoav$ can be inhomogeneous even under
average over positions of impurities. The key observation is that rarefied
bands emerge from irregular orbits of the action $\Pens$ of the
translation group $C_n$ on the set $\Phiens$ of all magnetic
configurations of the ensemble of magnets.

Let $\co\subset\Phiens$ be an orbit with
\be
|\co|=\bar{\kappa}=n/\kappa
\ee
elements, i.e.\ with the stabilizer $C_{\kappa}\lhd C_n$. Then the
restriction of $\Pens$ to this orbit decomposes as
\be
\Pens|_{\mbox{\rm lc}_{\mbox{\scriptsize \mathc}}{\textstyle \co}}\cong
\bigoplus_{k\in B|\kappa}\Gamma_k,
\ee
where the set $B|\kappa$ is the $\kappa$-tuply rarefied Brillouin zone (cf.\
Eq.\ (\ref{rband})). In particular, for a regular orbit we have $\kappa=1$,
so that
\be
|\co_{\rm reg}|=n
\ee
and
\be
\Pens|_{\mbox{\rm lc}_{\mbox{\scriptsize \mathc}}{\textstyle \co_{\rm reg}}}
\cong\bigoplus_{k\in B}\Gamma_k,
\ee
since $B|1=B$. Thus each regular orbit yields a full band. For $\kappa>1$ we
have 
\be
|B|\kappa|<n,
\ee
so that each regular orbit yields a rarefied band, and thus inhomogeneity of
the distribution $\rhoav$.

In the case of two impurities, i.e.\ $n'=2$, we have
\be
(S_{n-2}\times S_2)\cap C_n=\left\{
   \begin{array}{ll}
 C_2\;\;\;& \mbox{\rm for antipodal impurities ($n$ even, $S_2\subset C_n$),}\\
 C_1      & {\rm otherwise,}
   \end{array}\right.  
\ee
so that for the case of antipodal impurities the translational symmetry of
the magnet is broken only partially (from $C_n$ to $C_2$).

E.g.\ for $n=6$ and $n'=2$, with the host spin $s=1/2$ ($m=2$) and the
impurity spin $s'=1$ ($m=3$), we get
\be
|\ul2^{\ul4}\times\ul3^{\ul2}|=2^4\cdot 3^2=16\cdot 9=144
\ee
magnetic configurations for a given chemical configuration, i.e.\ for a
given distribution of the two impurities. The total number of magnetic
configurations for the ensemble is
\be
|\Phiens|=6\cdot 144=864.
\ee
When the translational symmetry is totally broken, we get a homogeneous
distribution. In the case of antipodal impurities the action $\Pens$ yields

\begin{center}
\begin{tabular}{rlr@{\hspace{1ex}configurations}}
138 & regular orbits of $C_6$, i.e.   & 828 \\
 12 & irregular orbits of $C_6$, i.e. &  36 \\ \cline{1-1}\cline{3-3}
150 & orbits                          & 864 \\
\end{tabular}
\end{center}

\begin{table}
\caption{Distribution of states for $n=6$, $n'=2$}
\begin{center}
\begin{tabular}{|l|rrrr|}\hline
               & \mcc0 & \mcc1 & \mcc2 & \mcd3 \\ \hline
full bands     &   138 &   138 &   138 &   138 \\
rarefied bands &    12 &   --- &    12 &   --- \\
$\rhoav$       &    25 &    23 &    25 &    23 \\ \hline
\end{tabular}
\end{center}
\label{tab1}
\end{table}


The Brillouin zone decomposes into generalized stars as 
\be
B=\{0\},\{\pm1\},\{\pm2\},\{\pm3\}.
\ee
The resulting distribution of states is given in Table \ref{tab1}. Rarefied
bands emerge from irregular orbits corresponding to the rarefied Brillouin zone
\be
B|2=\{0\},\{\pm2\}.
\ee

\section{Conclusions}
We have found that impurities break the obvious symmetry of the recipe of
Weyl, but preserve the hidden symmetry. As a result, the distribution of
quantum states over the Brillouin zone does not loose the hidden symmetry. In
particular, it remains constant on each generalized star. Moreover, the
procedure of averaging over the ensemble of magnets with translationally
equivalent distributions of impurities wipes out, partially or totally,
inhomogeneities of thus distribution. Rarefied bands emerge as a result of
irregular orbits of translation group action on the set of all magnetic
configurations of an ensemble of magnets.

\begin{thebibliography}{9}
\bibitem{ref1} H. Weyl, {\it Symmetry}, Princeton U.P., New Jersey 1952, pp. 
119--145.

\bibitem{ref2} M. Mucha, {\it Hidden Symmetries and Weyl's recipe}, in: W.
Florek, T. Lulek, and M. Mucha (Eds.), {\it Symmetry and Structural
Properties of Condensed Matter}, World Sci., Singapore 1991, p. 19.
  
\bibitem{ref3} 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{ref4} B. Lulek, {\it Rarefied bands within the Heisenberg model of
magnetism}, in: W. Florek, T. Lulek, and M. Mucha (Eds.), {\it Symmetry and
Structural Properties of Condensed Matter}, World Sci., Singapore 1991, p.
131.

\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} M. Ku\'zma, {\it Fractal symmetries of line polymers}, in: W.
Florek, T. Lulek, and M. Mucha (Eds.), {\it Symmetry and Structural
Properties of Condensed Matter}, World Sci., Singapore 1991, p.  379.
 
\bibitem{ref7} T. Lulek, {\it Density of states in the reciprocal lattice for
a one-dimensional periodic Heisenberg magnet}, J. Physique {\bf 45} (1984) 29.

\bibitem{ref8} H.O. Foulkes, {\it Characters of symmetric groups induced by
characters of cyclic subgroups}, Proc. Conf. Combinatorial Mathematics,
Mathematical Institute, Oxford 1972, p. 141.

\bibitem{ref9} 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.
\end{thebibliography}

\end{document}

