Spiral orientational order in quantum Hall skyrmion lattices
Sumithra Sankararaman\cite{SS} and R. Shankar\cite{RS}
We investigate the existence of spiral ordering in the planar spin orientation of skyrmions localised on a face centered rectangular lattice (FCRL). We use the non-linear sigma model (NLSM) to numerically calculate the minimum energy configurations of this lattice around the $\nu =1$ quantum Hall ground state. Our variational ansatz contains an angle $\theta$, characterising the FCRL and an angle $q$, characterising the orientational order. As $\nu$ is increased towards one, there is a smooth transition from the triangular lattice (TL) characterised by ($\theta,q$)= ($120^o,120^o$) to FCRLs with spiral orientational order. The novel feature we find is that these phases are characterised by ($\theta,q$) values such that $\theta+q=240^o$ (same as the TL phase). As $\nu$ increases further towards one, there is a sharp transition from the FCRLs to the square lattice (SL) characterised by ($\theta,q$)= ($90^o,180^o$). Consequently, the parameter $\theta+q$ jumps sharply at the FCRL-SL transition and can serve as an order parameter to characterise it.
merge.tex
\documentstyle[twocolumn,aps,epsf]{revtex}
\def\myfigure#1#2{{\leftskip=0.000753\textwidth \rightskip\leftskip\small
\begin{figure}\baselineskip=14pt plus 2pt minus 1pt
\centerline{#1}\nobreak\smallskip\nobreak #2\end{figure}}}
\global\firstfigfalse
\draft
\newcommand{\vn}{{\bf n}}
\newcommand{\vx}{{\bf x}}
\newcommand{\vy}{{\bf y}}
\newcommand{\vR}{{\bf{R}}}
\newcommand{\vq}{{\bf q}}
\newcommand{\vS}{{\bf{S}}}
\newcommand{\vG}{{\bf{G}}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\nn}{\nonumber}
\newcommand{\ra}{\rightarrow}
\begin{document}

\title{Spiral orientational order in quantum Hall skyrmion lattices}
\author{Sumithra Sankararaman\cite{SS} and R. Shankar\cite{RS}}
\address{Institute of Mathematical Sciences, CIT Campus, Chennai 600
113, India.}
\date{\today}
\maketitle

\begin{abstract}
We investigate the existence of spiral ordering in the planar spin
orientation of skyrmions localised on a face centered rectangular 
lattice (FCRL).  We use the non-linear sigma model (NLSM) to numerically 
calculate the minimum energy configurations of this lattice  around the 
$\nu =1$ quantum Hall ground state. Our variational ansatz contains an 
angle $\theta$, characterising the FCRL and an angle $q$, 
characterising the orientational order. As $\nu$ is increased towards one,
there is a smooth transition from the triangular lattice (TL) characterised 
by ($\theta,q$)= ($120^o,120^o$) to FCRLs with spiral orientational order. 
The novel feature we find is that these phases are characterised by 
($\theta,q$) values such that $\theta+q=240^o$ (same as the TL phase). 
As $\nu$ increases further towards one, there is a sharp transition from the
FCRLs to the square lattice (SL) characterised by ($\theta,q$)= ($90^o,180^o$). 
Consequently, the parameter $\theta+q$ jumps sharply at the FCRL-SL
transition and can serve as an order parameter to characterise it.
\end{abstract}
\bigskip
The lowest energy charged excitations about the $\nu=1$ ferromagnetic
quantum Hall ground state are skyrmions. They are topological objects
in which the spin gradually twists over an extended region. Their spin 
is greater than $1/2$ and they carry an electric charge of
$\pm e$. Skyrmions have a topological charge which equals their electric
charge at $\nu=1$ \cite{sondhi,leemac,poly}. Skyrmionic excitations 
are favoured over single particle excitations when the Land\'e g-factor 
is small i.e $g \ra 0$ \cite{sondhi,fertig1,fertig2}. 
Skyrmions which are produced around the $\nu=1$ ground state have been 
experimentally seen by in OPNMR and optical magneto-absorption experiments 
on Ga nuclei in an electron doped multiple quantum well structure
\cite{barrett,aifer}. 
They see a sharp fall in the spin polarisation of the 2D electron gas 
on either side of $\nu=1$, which indicates that a number of spins are 
being flipped by the addition or removal of a single electron, in 
contrast to single particle excitations where no extra spin is flipped. 
Low $g$ experiments observe large skyrmions with a large number of 
flipped spins \cite{shukla,nick}.

The low temperature ground state of a system of interacting skyrmions
is expected to be a crystalline lattice \cite {brey,green,madan,timm}. 
Earlier calculations performed in the mean field limit by Brey {\it et. al.} 
\cite{brey} suggest  that the ground state of a dilute system 
of a skyrmions is a square Skyrme crystal with an anti-ferromagnetic order 
in the planar component of the spin of the skyrmion. Later studies by Green 
{\it et. al.} , Rao {\it et. al.} and Timm {\it et. al.} analyse the 
classical ground state of the skyrmion lattice as $\nu \ra 1$ using the 
non-linear sigma model (NLSM) \cite{green,madan,timm}. Green {\it et. al} 
conclude that the ground state is a triangular lattice with N\'eel ordering 
\cite{green}.  Rao {\it et. al.} report sharp $triangle \ra square \ra 
triangle$ transitions in the skyrmion lattice as a function of the 
filling factor near $\nu = 1$ and at T=0 \cite{madan}. Timm {\it et. al.} 
study a system of well separated skyrmions as described by an 
anti-ferromagnetic XY model and propose a T=0 phase diagram in which the 
TL and SL phases are separated by N\'eel ordered, centered 
rectangular phases \cite{timm}. The classical and quantum phase transitions
occurring in the Skyrme crystal has been studied using Hartree-Fock
calculations by C\^ot\'e {\it et. al.} \cite{stoof}.  

In this paper, we consider spiral ground states as candidate minimum
energy solutions of a system of skyrmions described by an effective 
classical O(3) NLSM.  Spiral ground states are known to be the ground 
states of frustrated anti-ferromagnets.  For example, the triangular 
lattice with ABC sublattice spin ordering is a frustrated spin system 
in which the spin at every vertex is rotated through $120^o$ relative 
to the others. 

Our calculations are is valid for a system of overlapping skyrmions
at small $g$. The Coulomb and Zeeman terms in the
NLSM compete to generate a size for the skyrmion and to drive the
TL-SL transition through the FCRLs.  This is in contrast
to the regime of Timm {\it et. al.} where the Zeeman contribution is 
neglected in the large separation limit \cite{timm}. The FCRLs they 
observe do not show spiral ordering of spins.

We find that as $\nu$ increases towards $1$, the system changes 
(at non-zero $g$) smoothly from the TL phase characterised by 
($\theta,q$)= ($120^o,120^o$) to
FCRLs with spiral orientational order. The FCRLs have different 
values of $\theta$ and $q$, but their $\theta+q$ value is always $240^o$, 
which is the value of $\theta+q$ in the TL phase. As $\nu$ is increased 
further there is another sharp transition from the FCRLs to the SL 
phase characterised by ($\theta,q$)=($90^o,180^o$). The parameter $\theta+q$ 
shows a sharp jump from $240^o$ to $270^o$ at the FCRL-SL transition. 
Hence it is a convenient order parameter for this transition. The spin 
polarisation of the system  varies smoothly with $\nu$ and does not 
show an abrupt behaviour at any of the two transitions. The FCRL phases 
seem to smoothen out the jump in the spin polarisation, which was observed 
at the transition to the SL phase in previous work \cite{madan}. 

We consider skyrmionic excitations about the $\nu=1$ ground state. The
number density of skyrmions is given by $n_{sky} = \frac{1-\nu}{\nu}n_c$,
where $n_c$ is the carrier density. In our calculations we use the
carrier density, $n_c = 1.5 \times 10^{11}~cm^{-2}$ and change $\nu$ by
tilting the magnetic field. 
We localise the skyrmion centers on the lattice points of
a FCRL shown in Fig.\ 1. 
$\theta$ is the angle between the Bravais lattice vectors and $e$ is the 
length of the Bravais lattice basis vectors, which vary with the 
filling factor. We choose a skyrmion density ($n_{sky}$) of 1 skyrmion
per unit cell (i.e one skyrmion per lattice point). 

We measure all lengths in units of the magnetic length $l_{c}=\sqrt
\frac {\hbar} {e B}$ and all energies in terms of the cyclotron
energy $\hbar \omega_c$,
where $\omega_c = (eB/m^*c)$, $m^*$ is the effective mass of the
electron.
The area of the unit cell is fixed by the filling factor 
and is given by 
\be
A = \frac {1}{n_{sky}}=e^2 \sin\theta = \frac{2\pi}{1-\nu}
\ee

The local spin polarisation which is represented by unit vector field
$\vn(\vx)$ is stereographically projected onto the complex plane
by the transformation
 $w=\cot(\theta/2) e^{i\phi}$, where $\theta$ and $\phi$ are the polar 
angles of the spin vector $\vn (\vx)$. In the rest of the paper we will
work with the planar spin variable $w$. 
 
The topological charge density is given by
\bea
\rho(\vx) &=& \frac{1}{4\pi}\epsilon_{ij}\vn.(\partial_i \vn \times 
\partial_j \vn) \nn \\
&=& \frac {\epsilon_{ij}
\partial_i w \partial_j \overline w} {2 \pi i (1+w \overline w)^{2}}
\eea
The topological charge, 
$Q(\vx)=\int_{\Box} d^{2}x~\rho(\vx) = 1$
($\int_{\Box}$ denotes integration over the unit cell).

The low energy, long wavelength excitations
about the $\nu=1$ ground state are accurately described by the NLSM. 
The NLSM energy functional has to be
minimised for different filling factors to get the minimum energy
configurations of this lattice.
The NLSM energy functional with Zeeman and Coulomb interactions is 
\cite{sondhi,green,madan}:

\be
E = E_{grn}+  E_z + E_{coul}
\ee
The gradient or the spin exchange term proportional to $\int d^2x
{\mid \partial_i \vn(\vx) \mid }^2,(i = x,y)$, is
calculated as 
\be
\label{grn}
E_{grn} = \frac{\gamma}{2} \int_{\Box} d^{2}x \frac {(\partial_x w 
\partial_x \overline w
+\partial_y w \partial_y \overline w )} {(1 + w \overline w)^2}
\ee
where
$\gamma = \frac {e^*}{16 \sqrt{2 \pi}}$.
The gradient energy density is {\it $ E_{grn}$ /A} , where A is
the area of the unit cell.

This term alone is the pure NLSM and it has scale invariant solutions
\cite{poly}.

The Zeeman term is proportional to the z-component of the
total spin i.e. to
$\frac{\nu}{2 \pi} \int_{\Box} d^2x
\frac {(1 + n^z)}{2}$ (where $n^z$ is the z-component of $\vn(\vx)$).
In our units the average number of electrons is $\frac{\nu}{2
\pi}$. 
The z-component of the total spin is 
\bea
(Total~spin)_z =  \frac{\nu}{2 \pi}\int_{\Box} d^{2}x \frac 
{\overline w w}{(1 + \overline w w)}
\eea
Therefore,
\bea
\label {ezee}
E_z = g^* \frac{\nu}{2 \pi}\int_{\Box} d^{2}x \frac {\overline w w}
{(1 + \overline w w)}
\eea
where $g^{*} =\frac {g \mu_B B}{\hbar \omega_c}$.
The Zeeman energy density is therefore {\it $E_z/A$},where A is the 
area of the unit cell.

The Coulomb energy density term is a term of the form
\be
E_{coul} = \frac{e^*}{2}\frac {1}{A_{tot}} \int_{\vx,\vy} \rho(\vx)
\frac {1}{\mid \vx - \vy \mid} \rho(\vy)
\ee
where $e^{*} = (e^2/Kl_c)(1/\hbar \omega_c)$ and $A_{tot}$ is the 
total area of the lattice. 

The Coulomb term arises because the electric charge density is 
proportional to the topological charge density. Since the topological 
charge density explicitly appears in the above expression the 
spin orientation gets automatically tied to the Coulomb energy. 
The four dimensional integral in the Coulomb term can be converted 
to a sum over the reciprocal lattice.    
\be
{E_{coul} =  \frac {e^{*} \pi}{A^2} \sum_{\{\vG_R\}} {\mid \tilde{\rho}
(\vG_{R})\mid}^{2}
\frac {1} {\mid \vG_R \mid}}
\ee
where $\tilde{\rho} (\vG_{R}) = \int_{\vx \in \Box} \rho(\vx) e^{i \vG_{R}.
\vx}$, 
$\vG_R$ lies in the reciprocal lattice and
A is the area of the unit cell.

\begin{figure}
\myfigure{\epsfysize1.7in\epsfbox{fcenlat.eps}}{\vskip 0inFig.\ 1~~Face
centered rectangular lattice with basis vectors:~~{\bf{e1}}=$(e\cos\theta/2,
-e\sin\theta/2)$,{\bf{e2}}=$(e\cos\theta/2,e\sin\theta/2)$,
nn,nnn,nnnn indicate nearest neighbour,next nearest neighbour 
and third nearest neighbour respectively. The box is a unit cell.}
\end{figure}

We use the following ansatz to minimise the energy functional:
\be
\label{ansatz}
w(z) = \sum_{\{\vR\}}\frac {\lambda e^{i\vq.\vR}} {z - \vR}
\ee
where, $\lambda$ sets the scale for the skyrmion size,
$\vq.\vR$ is the spiral angle.  The angle $\theta$ characterises the lattice.
This analytic ansatz exactly minimises the gradient term and gives scale
invariant solutions \cite{poly}. For analytic skyrmions the gradient 
term does not influence positional or spin-orientational ordering.
For overlapping skyrmions at low $g$, we still use
the same ansatz since their size is large and the exponential damping
term used in ref.\ \cite{madan} does not have any significant effect. 
The size is determined by the competition between the Zeeman and Coulomb terms.
We find that the Zeeman term prefers the SL phase and the Coulomb term 
prefers the TL phase. We expect to see FCRLs with spiral ordering 
when the transition occurs between the above phases.  The NLSM 
energy functional has to be minimised with respect to the four 
variational parameters $\lambda, \theta,\vq$(two components). 
      
To get an idea of the kind of positional and magnetic ordering expected 
in this system we look at a toy model , a spin system described by 
an XY model on the FCRL shown in the figure. 
The free energy of the system is  : 
\be
E = \sum_{(i,j)} J_{ij} \vS_i.\vS_j
\ee
We look at position and magnetic configurations which lie between the
TL and SL phases, which are FCRLs.
We consider only nearest neighbour ($J_1$), next nearest neighbour
($J_2$) and third nearest neighbour ($J_3$) couplings on the
lattice. These configurations have 4 nearest neighbours and 6 next nearest 
neighbours (except at the square and triangle end points).
We choose a spiral state with spiral angle $\vq . \vR = q_x R_x + q_y R_y$,
where $R_x$ and $R_y$ are co-ordinates of a lattice point and $q_x$ and
$q_y$ are the angles through which the spin turns for every jump to a
lattice point. For the above configurations (except exactly at square
and triangle end points), the free energy has the form :
\bea
E = 2J_1 \cos(q_x) + 2J_1 \cos(q_y) + 2J_2 \cos(q_x+q_y) \nn \\
-2J_3 \cos(q_x-q_y)
\eea
This free energy is minimised with respect to $q_x$ and $q_y$. 
In the relevant parameter ranges, we find the solutions to be of the
form $(q_x,q_y) = (q,q)$. This motivates the
use of $(q_x,q_y) = (q,q)$ in the exponential term of the 
ansatz in Eq.\ (\ref {ansatz}).

With the above combination the number of variational parameters has
reduced to three $(\theta,\lambda,q)$.  The variational ansatz given 
above is used in the NLSM energy functional and all energies are 
calculated and minimised numerically (to 1 part in $10^5$)
with respect to these parameters . The filling factor $\nu$ is
varied for different values of the gyromagnetic ratio $g$ and the phase
diagram for the system in the $\nu-g/g_0$ plane is obtained ($g_0$ is
the physical value of the Land\'e g-factor of the system). 

The accuracy of the routine was tested by integrating the 
topological charge density over a unit cell to get the topological charge, 
which should be 1 since there is one skyrmion per unit cell. 
An accuracy of $1$ part in $10^6$ was noted in the calculation of
topological charge.

Another test of accuracy can be done at $g=0$. The only energy in 
the system is the Coulomb energy and this sets the length scale 
in the problem. It can then be shown that ($\theta,q$) should be 
independent of $\nu$ and that $\lambda \propto e$.  
We find that the configuration which favors minimum Coulomb 
energy is the TL phase, ($\theta ,q $)=($120^o,120^o$) and verify that
$\lambda \propto e$.

\begin{figure}
\myfigure{\epsfysize2.5in\epsfbox{phase.eps}}{\vskip 0inFig.\ 2~~Phase
diagram in the $\nu-g/g_0$ plane} 
\end{figure}
\begin{figure}
\myfigure{\epsfysize3.0in\epsfbox{thetaq.eps}}
{\vskip 0inFig.\ 3~~Plots
of $\theta+q-\nu$ values at different $g/g_0$. The solid lines are the
$\theta+q$ values at the TL and SL, the dots indicate the numerical
values obtained. Note the jump in the vicinity of the square transition.} 
\end{figure}

The minimum energy configurations were found for different filling factors at
various $g$ values. For $g=0$ there is no transition from the TL
phase as the filling factor is brought towards one.  
The TL-FCRLs transition starts appearing at non-zero $g$. The
second transition to the SL phase, ($\theta,q$)=($90^o,180^o$), occurs 
very close to $\nu=1$ when $g=0.1g_0$. Both the transitions shifted 
away from $\nu=1$ as the $g$ value was increased and the intermediate 
region of FCRLs broadens. The phase diagram of the skyrmion lattice 
in the $\nu-g/g_0$ plane is shown in Fig.\ 2.  

The FCRLs appear as intermediate phases in regions of the phase diagram 
where the NLSM is a good description of the quantum Hall system (i.e low $g$
and high $\nu$ regions). The $\theta+q-\nu$ diagram shown in Fig.\ 3. 
The $\theta+q$ value is at $240^o$ at the TL and FCRLs and is at $270^o$ 
in the SL. The sharp jump seen in $\theta+q$ occurs at all non-zero $g$ values. 
Hence, we propose $\theta+q$ as an order parameter characterising the 
FCRL-SL transition. 

The spin polarisation (sp) of the system of skyrmions at a particular
$\nu$ is
\bea
sp&=& \frac{<s_z>}{N_c} = \frac{1}{N_c}\int s_z d^2 x 
 \nn \\
&=& \frac{1-\nu}{\nu}(Total~Spin)_z - \frac{1}{2}  
\eea
The FCRLs seem to smoothen out the jump in the spin polarisation that
was observed in ref.\ \cite{madan}. The value of total spin at $g=0.1g_0$
was found to be 27 at $\nu=0.99$. This means that many spins flip and
explains the negative value of spin polarisation . Direct comparison
of this value with those obtained in low $g$ experiments \cite{shukla,nick}
is not possible since we measure total spin at non-zero 
skyrmion density whereas the experiments measure spin for a thermally 
activated skyrmion-anti-skyrmion pair at $\nu=1$.

\begin{figure}
\myfigure{\epsfysize2.5in\epsfbox{spinpol.eps}}{\vskip 0inFig.\ 4~~Spin
Polarisation at different $g/g_0$ indicated in box brackets.}
\end{figure}
Novel FCRL phases with spiral ordering such that $\theta+q = 240^o$ appear 
in the low $g$ limit (overlapping skyrmions). The TL-FCRL transition is smooth
characterised by $\theta+q = 240^o$. The second FCRL-SL transition is sharp 
and $\theta+q$ jumps from $240^o$ to $270^o$, making this a convenient
order parameter for this transition. There are no spiral phases with
$\theta+q$ between these angles.  

Our calculations do not explain why FCRLs with these values 
of $(\theta,q)$ occur
but think that this is due to the hedgehog nature of the skyrmion. 
We have minimised the two-body energy 
functional of Timm {\it et. al.} and find that FCRLs with a general 
spiral ordering of spins do not occur. We confirm their phase diagram 
for a system of well separated skyrmions , which shows TL-FCRLs 
(N\'eel ordered)-SL transitions \cite{timm}. In their calculation, 
the Coulomb energy which favours the TL and the exchange energy which 
favours the SL, compete to drive these transitions.  We conclude that the 
spiral ordering we observe is not a two-body effect and occurs because 
of three-body or higher order effects. 

We thank Madan Rao and Surajit Sengupta for helpful discussions.
  
\begin{references}

\bibitem[*]{SS} sumithra@imsc.ernet.in

\bibitem[**]{RS} shankar@imsc.ernet.in

\bibitem{sondhi}
S. L. Sondhi, A. Karlhede, S. A. Kivelson and E. H. Rezayi, Phys. Rev. 
{\bf B47}, 16419 (1993).
 
\bibitem{leemac}
D. H. Lee and C. L. Kane, Phys. Rev. Lett. {\bf 64}, 1313 (1990)\,; 
A. H. MacDonald, cond-mat/9601144

\bibitem{poly}
A. A. Belavin and A. M. Polyakov, JETP Lett. {\bf 22}, 245 (1975).

\bibitem{fertig1}
H. A. Fertig, L. Brey, R. C\^ot\'e and A. H. MacDonald,  Phys. Rev.
{\bf B50}, 11018 (1994)\,; K. Moon, H. Mori, K. Yang, S. M.
Girvin, A. H. MacDonald, L. Zheng, D. Yoshioka and S. C.
Zhang, {\it ibid.} {\bf 51}, 5138 (1995).

\bibitem{fertig2}
H. A. Fertig, Luis Brey, R. C\^ot\'e, A. H. Macdonald, A. Karlhede and
S. L. Sondhi, Phys. Rev. {\bf B55}, 10671 (1997). 

\bibitem{brey}
L. Brey, H.A. Fertig, R.C\^ot\'e and A. H. Macdonald, Phys. Rev. Lett
{\bf 75}, 2562 (1995).

\bibitem{green}
A. G. Green, I.I. Kogan and A. M. Tsevelik, Phys. Rev. {\bf B54}, 
16398 (1996).

\bibitem{madan}
Madan Rao, Surajit Sengupta and R. Shankar, Phys. Rev. Lett {\bf 79},
3998 (1997).  

\bibitem{timm}
Carsten Timm, S. M. Girvin and H. A. Fertig, Phys. Rev. {\bf B58}, 10634
(1998). 

\bibitem{stoof}
R. C\^ot\'e, A. H. MacDonald, L. Brey, H. A. Fertig, S. M. Girvin and H.
T. C. Stoof, Phys. Rev. Lett. {\bf 78}, 4825 (1995).
 
\bibitem{barrett}
S. E. Barrett, G. Dabbagh, L. N. Pfeiffer, K. W. West and R.
Tycko, Phys. Rev. Lett. {\bf 74}, 5112 (1995)\,; S. E. Barrett,
G. Dabbagh, L. N. Pfeiffer, K. W. West, and R. Tycko, Phys. Rev.
Lett. {\bf 72}, 1368 (1994)\,; R. Tycko, S. E. Barrett, G. Dabbagh,
L. N. Pfeiffer and K. W. West, Science {\bf 268}, 1460 (1995).

\bibitem{aifer}
E. H. Aifer, B. B. Goldberg and D. A. Broido, Phys. Rev. Lett.
{\bf 76}, 680 (1996).

\bibitem{shukla}
S. P. Shukla,M. Shayagan,S. P. Parihar, S. A. Lyon, N. R. Cooper and A.
A. Kiselev, Phys. Rev. {\bf B61}, 4469 (2000).

\bibitem{nick}
D. R. Leadley, R. J. Nicholas, D. K. Maude, A. N. Utjuzh, J. C. Portal,
J. J. Harris and C. T. Foxon, Semicond. Sci. Technol. 13, 671
(1998).
\end{references}
\end{document}

cond-mat0101451.extracted_bib
\bibitem[*]{SS} sumithra@imsc.ernet.in


\bibitem[**]{RS} shankar@imsc.ernet.in


\bibitem{sondhi}
S. L. Sondhi, A. Karlhede, S. A. Kivelson and E. H. Rezayi, Phys. Rev. 
{\bf B47}, 16419 (1993).
 

\bibitem{leemac}
D. H. Lee and C. L. Kane, Phys. Rev. Lett. {\bf 64}, 1313 (1990)\,; 
A. H. MacDonald, cond-mat/9601144


\bibitem{poly}
A. A. Belavin and A. M. Polyakov, JETP Lett. {\bf 22}, 245 (1975).


\bibitem{fertig1}
H. A. Fertig, L. Brey, R. C\^ot\'e and A. H. MacDonald,  Phys. Rev.
{\bf B50}, 11018 (1994)\,; K. Moon, H. Mori, K. Yang, S. M.
Girvin, A. H. MacDonald, L. Zheng, D. Yoshioka and S. C.
Zhang, {\it ibid.} {\bf 51}, 5138 (1995).


\bibitem{fertig2}
H. A. Fertig, Luis Brey, R. C\^ot\'e, A. H. Macdonald, A. Karlhede and
S. L. Sondhi, Phys. Rev. {\bf B55}, 10671 (1997). 


\bibitem{brey}
L. Brey, H.A. Fertig, R.C\^ot\'e and A. H. Macdonald, Phys. Rev. Lett
{\bf 75}, 2562 (1995).


\bibitem{green}
A. G. Green, I.I. Kogan and A. M. Tsevelik, Phys. Rev. {\bf B54}, 
16398 (1996).


\bibitem{madan}
Madan Rao, Surajit Sengupta and R. Shankar, Phys. Rev. Lett {\bf 79},
3998 (1997).  


\bibitem{timm}
Carsten Timm, S. M. Girvin and H. A. Fertig, Phys. Rev. {\bf B58}, 10634
(1998). 


\bibitem{stoof}
R. C\^ot\'e, A. H. MacDonald, L. Brey, H. A. Fertig, S. M. Girvin and H.
T. C. Stoof, Phys. Rev. Lett. {\bf 78}, 4825 (1995).
 

\bibitem{barrett}
S. E. Barrett, G. Dabbagh, L. N. Pfeiffer, K. W. West and R.
Tycko, Phys. Rev. Lett. {\bf 74}, 5112 (1995)\,; S. E. Barrett,
G. Dabbagh, L. N. Pfeiffer, K. W. West, and R. Tycko, Phys. Rev.
Lett. {\bf 72}, 1368 (1994)\,; R. Tycko, S. E. Barrett, G. Dabbagh,
L. N. Pfeiffer and K. W. West, Science {\bf 268}, 1460 (1995).


\bibitem{aifer}
E. H. Aifer, B. B. Goldberg and D. A. Broido, Phys. Rev. Lett.
{\bf 76}, 680 (1996).


\bibitem{shukla}
S. P. Shukla,M. Shayagan,S. P. Parihar, S. A. Lyon, N. R. Cooper and A.
A. Kiselev, Phys. Rev. {\bf B61}, 4469 (2000).


\bibitem{nick}
D. R. Leadley, R. J. Nicholas, D. K. Maude, A. N. Utjuzh, J. C. Portal,
J. J. Harris and C. T. Foxon, Semicond. Sci. Technol. 13, 671
(1998).
