Previously we have indicated the relationship between quantum groups and strings via WZWN models. In this note we discuss this relationship further and point out its possible applications to cuprates and related materials. The connection between quantum groups and strings is one way of seeing the validity of our previous conjecture [i.e. that a theory for cuprates may be constructed on the basis of quantum groups]. The cuprates seems to exhibit statistics, dimensionality and phase transitions in novel ways. The nature of excitations [i.e. quasiparticle or collective] must be understood. The Hubbard model captures some of the behaviour of the phase transitions in these materials. On the other hand the phases such as stripes in these materials bear relationship to quantum group or string-like solutions. One thus expects that the relevant solutions of Hubbard model may thus be written in terms of stringy solutions. In short this approach may lead %ins by me to the non-perturbative formualtion of Hubbard and other condensed matter Hamiltonians. %%%%%%%%%%%%%%%% The question arises that how a 1-d based symmetry such as quantum groups can be relevant in describing a 3-d [spatial dimensions] system such as cuprates. The answer lies in the key observation that strings which are 1-d objects can be used to describe physics in $d$ dimensions. For example gravity [which is a 3-d [spatial] plus time] phenomenon can be understood in terms of 1-d strings. Thus we expect that 1-d quantum group object induces physics in 2-d and 3-d which may be relevant to the cuprates. % Ins collective excitations % Following our recent conjecture to model the phenomenona of % antiferromagnetism and superconductivity by quantum symmetry % groups, we discuss in the present note how to construct % a workable scenario using this symmetry. In particular % we propose to patch the relevant critical points. This means % we identify fixed points, corresponding to various $k$ % or $q$ [since the two are related] and make expansion % around these points, to control these expansion we % can impose gauge structure and we thus arrive at quantum % group based gauge theory or collection of classical gauge % theories which represent the condensed matter system such as % cuprates. This is different than ordinary gauge theories % in several ways, for in ordinary field theory one has % well-defined critical point here the critical points are % not unique or simple. In short the real transition than in % condensed matter system is represented by collection or % {ensemble average} of the several chosen critical points % which come from ordinary field theory [gauge theory]. % This idea may reveal the connection between % Hubbard model and gauge theory and string theory. % In short it can lead %%ins by me % to the non-perturbative formualtion of Hubbard and % other condensed matter Hamiltonians. % This choice is also motivated to accomodate % the several competing phases which are or may be present in these % and related materials, such as stripe phase [mesoscopically % ordered phase], Luttinger liquids, nearly antiferromagnetic % Fermi liquids, charge-ordered Fermi liquids, glassy phase, % and perhaps more. The existence and the behavior of pseudo-gap and % lattice distortion are also an important consideration. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Possible motivations and rationale for these choices are % outlined. One of the prime motivations underlying our proposal % is the experimental observation of {\em stripe} structure [phase] % in high T$_{c}$ superconductivity [HTSC] materials. A number % of experimental techniques have recently observed that the % ${CuO_{2}}$ are rather inhomogeneous, providing evidence for % phase separation into a two component system. i.e. carrier-rich % and carrier-poor regions. % In particular, extended x-ray absorption fine structure [EXAFS] % demonstrated that these domains forms stripes of undistorted and % distorted local structures alternating with mesoscopic length scale % comparable with coherence length in HTSC. %
cond-mat0102097.tex
%Ins
% dated 9/4/99
% 9-8-2000
% Email [electronic copy] also sent to pla@elsevier.nl
% L.J.Sham, email lusham@ucsd.edu
%Subject: Submit ``Quantum Group based Modeling for the description
%of high temperature superconductivity and antiferromagnetism''
% PLA@ELSEVIER.NL
%\documentstyle[12pt]{letter}
%\begin{document}
%\address{Theory Group\\
%         KEK\\
%         Oho 1-1, Tsukuba-shi\\
%         Ibaraki 305, Japan}
%\begin{letter}{Editors \\
%              Editorial Office of Physics Letter A \\
%              Elsevier Science B.V.,Physics Letters A\\
%              P.O. Box 2759, 1000 CT Amsterdam, Netherlands}
%\opening{Dear Sir;}
%Enclosed please find a manuscript entitled  
%``The choice of the symmetry group:$SO(7)$''
%submitted for publication in Phys. Lett. A as 
%Its lanl archive number is cond-mat/004xxx

%The manuscript is prepared by REVTEX 

%Your consideration would be appreciated very much.
% Please acknowledge the receipt of this paper as soon as possible
%\hskip 2cm Sincerely yours,
%\vskip 0.5cm 
%\hskip 2cm S.Alam 
%\vspace{1.5cm}
%Email: sher@ccth2.kek.jp
%\end{letter}
%\end{document}

%%%%%%%%%%%%%%%%%%%Revtex File%%%%%%%%%%%%%
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\begin{document}
\draft
\hfill\vbox{\baselineskip14pt
            \hbox{\bf ETL-xx-xxx}
            \hbox{ETL Preprint 00-xxx}
%            \hbox{\today}}
%            \hbox{January 1998}}
            \hbox{March 2000}}
\baselineskip20pt
\vskip 0.2cm 
\begin{center}
{\Large\bf Quantum Groups, Strings and HTSC materials}
\end{center} 
\vskip 0.2cm 
\begin{center}
\large Sher~Alam$^{1}$,~M.~O.~Rahman$^{2}$,~T.~Yanagisawa$^{1}$,
and H.~Oyanagi$^{1}$
%\footnote{Permanent address: Department of Physics, University
%of Peshawar, Peshawar, NWFP, Pakistan.}
\end{center}
\begin{center}
$^{1}${\it Physical Science Division, ETL, Tsukuba, Ibaraki 305, Japan}\\
$^{2}${\it  GUAS \& Photon Factory, KEK, Tsukuba, Ibaraki 305, Japan}
%, S.~B.~Mohamed, and M.~O.~Rahman
%\footnote{GUAS \& Photon Factory, KEK, Tsukuba, Ibaraki 305, 
%Japan}
%\footnote{Permanent address: Department of Physics, University
%of Peshawar, Peshawar, NWFP, Pakistan.}
\end{center}
%\begin{center}
%{\it Physical Science Division, ETL, Tsukuba, Ibaraki 305, Japan}
%\end{center}
\vskip 0.2cm 
\begin{center} 
\large Abstract
\end{center}
\begin{center}
\begin{minipage}{14cm}
\baselineskip=18pt
\noindent
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This is the abstract
%\begin{abstract}
 Previously we have indicated the relationship between quantum groups and 
 strings via WZWN models. In this note we discuss this relationship 
 further and point out its possible applications to cuprates and related 
 materials. The connection between quantum groups and strings is one way 
 of seeing the validity of our previous conjecture [i.e. that a theory for 
 cuprates may be constructed on the basis of quantum groups].
 The cuprates seems to exhibit statistics, dimensionality and phase 
 transitions in novel ways. The nature of excitations [i.e. quasiparticle 
 or collective] must be understood. The Hubbard model captures some of 
 the behaviour of the phase transitions in these materials. On the other 
 hand the phases such as stripes in these materials bear relationship 
 to quantum group or string-like solutions. One thus expects that the 
 relevant solutions of Hubbard model may thus be written in terms of 
 stringy solutions. In short this approach may lead 
%ins by me
 to the non-perturbative formualtion of Hubbard and other condensed 
 matter Hamiltonians.  
%%%%%%%%%%%%%%%%
 The question arises that how a 1-d based symmetry such as quantum 
 groups can be relevant in describing a 3-d [spatial dimensions] system 
 such as cuprates. The answer lies in the key observation that strings 
 which are 1-d objects can be used to describe physics in $d$ dimensions. 
 For example gravity [which is a 3-d [spatial] plus time] phenomenon can 
 be understood in terms of 1-d strings. Thus we expect that 1-d quantum 
 group object induces physics in 2-d and 3-d which may be relevant to 
 the cuprates.

% Ins collective excitations

% Following our recent conjecture to model the phenomenona of
% antiferromagnetism and superconductivity by quantum symmetry
% groups, we discuss in the present note how to construct
% a workable scenario using this symmetry. In particular
% we propose to patch the relevant critical points. This means
% we identify fixed points, corresponding to various $k$
% or $q$ [since the two are related] and make expansion
% around these points, to control these expansion we 
% can impose gauge structure and we thus arrive at quantum
% group based gauge theory or collection of classical gauge 
% theories which represent the condensed matter system such as
% cuprates. This is different than ordinary gauge theories
% in several ways, for in ordinary field theory one has 
% well-defined critical point here the critical points are 
% not unique or simple. In short the real transition than in 
% condensed matter system is represented by collection or 
% {\it ensemble average} of the several chosen critical points 
% which come from ordinary field theory [gauge theory].
% This idea may reveal the connection between
% Hubbard model and gauge theory and string theory.
% In short it can lead 
%%ins by me
% to the non-perturbative formualtion of Hubbard and
% other condensed matter Hamiltonians.

% This choice is also motivated to accomodate
% the several competing phases which are or may be present in these
% and related materials, such as stripe phase [mesoscopically
% ordered phase], Luttinger liquids, nearly antiferromagnetic 
% Fermi liquids, charge-ordered Fermi liquids, glassy phase,
% and perhaps more. The existence and the behavior of pseudo-gap and
% lattice distortion are also an important consideration.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Possible motivations and rationale for these choices are 
% outlined. One of the prime motivations underlying our proposal
% is the experimental observation of {\em stripe} structure [phase]
% in high T$_{\rm c}$ superconductivity [HTSC] materials. A number 
% of experimental techniques have recently observed that the 
% ${\rm CuO_{2}}$ are rather inhomogeneous, providing evidence for 
% phase separation into a two component system. i.e. carrier-rich 
% and carrier-poor regions. 
% In particular, extended x-ray absorption fine structure [EXAFS] 
% demonstrated that these domains forms stripes of undistorted and 
% distorted local structures alternating with mesoscopic length scale
% comparable with coherence length in HTSC.
%\end{abstract}
\end{minipage}
\end{center}
\vfill
\baselineskip=20pt
\normalsize
\newpage
\setcounter{page}{2}
%------------------------------------------------------------------------ 
% Section: None
%\section{Introduction}+{others}, all in one!
%\twocolumn
% 20/6/2000	

	In a previous work one of us \cite{alam98} has advanced 
the conjecture that one should attempt to model the phenomena of
antiferromagnetism and superconductivity by using quantum
symmetry group. Following this conjecture to model the phenomenona of
antiferromagnetism and superconductivity by quantum symmetry
groups, three toy models were proposed \cite{alam99-1}, namely,
one based on ${\rm SO_{q}(3)}$ the other two constructed with
the ${\rm SO_{q}(4)}$ and ${\rm SO_{q}(5)}$ quantum groups. 
Possible motivations and rationale for these choices  
were outlined \cite{alam99-1}. In \cite{alam99-2} a model to 
describe quantum liquids in transition from 1d to 2d dimensional 
crossover using quantum groups was outlined. In \cite{alam00-1} 
the classical group ${\rm SO(7)}$ was proposed as a toy model to 
understand the connections between the competing phases and the 
phenomenon of psuedo-gap in High Temperature Superconducting Materials
[HTSC]. Then we proposed in \cite{alam00-2} an idea to
construct a theory based on patching critical points so as to
simulate the behavior of systems such as cuprates.
To illustrate our idea we considered an example 
discussed by Frahm et al., \cite{fra98}. The model
deals with antiferromagnetic spin-1 chain doped with
spin-1/2 carriers. In \cite{alam00-3} the connection between
Quantum Groups and 1-dimensional [1-d] structures such as
stripes was outlined. The main point of \cite{alam00-3}
is to emphasize that {\em 1-d structures play an important
role in determining the physical behaviour [such as the 
phases and types of phases these materials are capable of
exhibiting] of cuprates} and related materials.

	The question arises that how a 1-d based symmetry
such as quantum groups can be relevant in describing a
3-d [spatial dimensions] system such as cuprates. {\em The
answer lies in the key observation that strings which
are 1-d objects can be used to describe physics in
$d$ dimensions}. For example gravity \cite{kak91}
[which is a 3-d [spatial]
plus time] phenomenon can be understood in terms of 1-d
strings. Thus we expect that 1-d quantum group object
induces physics in 2-d and 3-d which may be relevant to
the cuprates. However we like to point out that
the notion of string is more general and does not
have to be restricted to quantum groups. The reason
for our choice are several. For one quantum groups
are intimately connected with WZWN models and
to strings. In turn WZWN based models are relevant
to disordered systems. The exact solution of Hubbard
model in 1-d is governed by quantum group symmetry.
In turn the Hubbard model captures much of the physics
of cuprates. We note that the t-J model and its
various generalizations are also advocated for the study
of cuprates. However the t-J model is but a limit of Hubbard
model [i.e. when $U/t \rightarrow \infty$]. Quantum
critical points may be naturally understood in terms
of quantum groups.

	Let us briefly comment on three different
general aspects of the cuprates and related materials:
\begin{itemize}
\item{}Statistics and fractionalization:-The cuprates seem 
to exhibit exotic statistics and charge fractionalization. 
Indications for electron fractionalization from Angle-Resolved 
Photoemission Spectroscaopy [ARPES] have
been reported. We have suggested to measure fractionalization
by using SET based experiments \cite{alam00-4}
\footnote{After about a week or so later of our submission a paper 
by Sentil and Fisher ``Detecting fractions of electrons in the 
high-$T_c$ cuprates''cond-mat/0011345, appeared without reference to 
our work. Some mathematical details were worked out but it used or 
was in line with basically our idea. What follows is the 
opinion of Sher Alam: I [Sher Alam] informed by email Sentil of this 
situation, in his arrogance he never replied. Such an attitude 
highly annoys me. We all use each other ideas in some form or 
the other, but more modest people have the courage to admit it. 
I hate pretentious and arrogant attitudes. Knowledge is God 
given right and not the exclusive right of some as they would 
have us believe it.}.
We note that the Fermi 
liquid is characterized by sharp fermionic
quasiparticle excitations and has a discontinuity in the
electron momentum distribution function. In contrast
the Luttinger liquid is characterized by charge $e$
spin $0$ bosons and spin $1/2$ charge $0$ and the
fermion is a composite of these [i.e. fractionalization].
It is well-known that transport properties
are defined via correlation functions.
The correlation functions of a Luttinger liquid
have a power law decays with exponents that
depend on the interaction parameters. Consequently
the transport properties of a Luttinger liquid
are very different from that of a Fermi liquid.
Photoemission experiments on Mott insulating oxides
seems to indicate the spinon and holon excitations
of a charge Luttinger liquid. However the experimental
signatures of Luttinger liquid are not totally
convincing. To this end we propose SET based experiments
to determine the Luttinger liquid behaviour of
the cuprates.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item{}Mixed dimensionality:- The cuprates and related
materials seem to exhibit mixed dimensionality. This
can be seen in many materials. We give an example of
Ca$_{2+x}$ Y$_{2-x}$ Cu$_{5}$ O$_{10}$ \cite{yam99}, a material 
which simple [i.e. simplicity in structural aspect and controlled
hole doping] and exhibits mixed dimensionality and can 
provide insight into the spin and charge dynamics of 
more complicated HTSC material. As a consequence of mixed 
dimensionality this material exhibits ferromagnetism and 
antiferromagnetism, introduction of holes leads to novel 
spin-charge dynamics in this magnetically frustrated system.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item{}Phases:-A variety of phases have been
reported in cuprates, for example, antiferromagnetic insulator,
superconducting, metallic, strange metal, spin-glass,
and insulating. 
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

	It has been known that the symmetry of Hubbard 
model is $SU(2)\times SU(2)/Z_2 \equiv SO(4)$, this
can be taken as a motivation to consider a gauge
model or NLSM for Hubbard based on $SU(2)$, $SU(2)\times$,
$SO(4)$ or even $Z_2$ or many combinations and
generalizations thereof depending on one's point
of view. We take the following point of view.
It is known that quantum groups in the context of
usual HH are tied to 1-d. This is used as an argument
against the application of quantum group to higher 
dimensions. We take the opposite attitude. Using the 
string notion instead of going from 1-d, to 2-d via
the usual route, we assume that fundamental
entities are 1-d objects [subject to quantum
group symmetry] i.e. strings, for example we introduce
the notion of Hubbard string. And attach the known
symmetery of Hubbard Hamiltonian i.e. $SU(2)\times 
SU(2)/Z_2 \equiv SO(4)$ for example as a Chan-Paton 
factor\footnote{Chan-Paton factor:-An open string has 
boundaries, i.e. endpoints, in quantum with distinguished 
endpoints it is natural to associate degrees of freedom
with these in addition to fields propagating in bulk.
Moreover it is natural and necessary to have simply
boundary conditions. For example in string theory
of strong interactions, one introduces SU(3)
flavor quantum numbers, the endpoints have 
quark-antiquark attached connected by `color-electric
flux tube}. We conjecture that the solution of 2-d
and 3-d Hubbard Hamiltonian can be written in
terms of the 1-d Hubbard string solutions.
This conjecture can be tested by looking at
specific examples. We thus have provided
a general framework for strongly correlated
electron systems, such as Quantum Hall systems
and HTSC materials and related materials. The recent 
data supports our earlier conjecture of important 
role of 1-d systems tied to quantum groups. 

P.W.Anderson and others have tried to prove spin charge
separation in 2-d. In our framework this is
not necessary and follows naturally as a 
consequence of the Hubbard string. It is
clear that spin charge separation takes
place on the 1-d Hubbard string, the dynamics
of the string leads to spin charge separation 
in 2-d and restricted confinement as appears
in the form of stripes observed in experiment.
The ARPES data shows a significant Fermi
surface with hotspots, this cannot be easily
explained in terms of Fermi Liquid or arranged
non Fermi Liquid behaviour. The collective
dynamics of 1-d strings can provide a suitable
explanation. For example one can interpret that 
the holes Fermi surface are the regions where
the collective or luttinger liquid-like behaviour 
dominates. Let us try to see this point intuitively,
string theory has several conservation laws,
which follows due to various symmetries. Now
some of these symmetries are broken in the target
space, for example in context of gravity conformal 
invariance is broken in the target space by
gravity and preserved on the 2-d surface
generated by string and yet it is the condition
of conformal invariance on string generated surface
that gives Einstein's equation coupled to a
scalar field in the target space. Thus in an
analogous manner one may have obtain a situation
where Luttinger liquid symmetry is preserved
in some regions of the Fermi surface and badly
broken in others. As the Hubbard strings fluctuate
they generate a Fermi surface where Luttinger 
correlations are preserved in only some regions.
The area or volume where non-Fermi liquid
persists is expected to be proportional to
the number of strings, which is turn is
determined by physical parameters such as
doping etc. A challenging problem is to define
precisely a measure for the ``correlations'' in 
the system. 

In short we expect that by formulating
the underling theory in terms of collective
excitations such as strings [which
are indicative of Luttinger liquids] we can
represent a general theory for strongly
correlated systems in 2-d and 3-d.
Yet another support for our point of view comes
from data of Uemura, where he finds 2-d bose
like behaviour, and these fit reasonably with
Berezinskii-Kosterlitz-Thouless [BKT] type of phase 
transitions. Within our framework this is natural since 
conformal invariance is intimately tied with strings. 
It would be interesting to look into the details
of the relationship between Uemura's data and the 
BKT-type phase transitions. 

We now comment briefly on the choice of Hamiltoinan
and naively on one kind of duality in this context.  
	As already mentioned and as is well-known the 
t-J model is a special limit of Hubbard model. Keeping
this point in mind we want to seek a non-trivial
unification of Hubbard Hamiltonian and the t-J model
using `duality' of couplings in a new way. 
To achieve this unification we start with
a small step. Let us explain. To this end let us first 
recall the usual argument based on perturbation theory 
which allows us to see the t-J model as the special limit of
the Hubbard Hamiltonian [HH]. The HH reads
\begin{eqnarray}
H = -t \sum_{<ij>\sigma} c_{i\sigma}^{\dagger}c_{j\sigma}
+ U  \sum_{i} n_{i~\uparrow}  n_{i~\downarrow}, 
\label{d1}
\end{eqnarray} 
where $<ij>$ means sum over nearest neighbours, i.e.
hybridization between neigbouring atoms. $c_{i\sigma}^{\dagger}$
($c_{i\sigma}$) are the creation (annihilation) operators
of electrons at site $i$ of spin $\sigma$ ($\sigma \equiv \uparrow
, \downarrow$).
An important limit of HH is half-filled and strong coupling
, i.e. $U \gg t$. 

The intermediate state has one doubly occupied atom
and the effective interaction [second-order] is simply
\begin{eqnarray}
H = t \sum_{<ij>\sigma^{'}} c_{i\sigma^{'}}^{\dagger}c_{j\sigma^{'}}
\frac{-1}{U}t\sum_{\sigma} c_{i\sigma}^{\dagger}c_{j\sigma}, 
\label{d2}
\end{eqnarray}
which is the Hamiltonian of antiferromagnetically coupled
spin-$\frac{1}{2}$ Heisenberg model with coupling $J=2t^{2}/U$ 
per bond. Naively one can regard in some sense 
the Hamitonian in Eq.~\ref{d2} as a `dual' with respect
to the coupling $U$ of the Hamiltonian in Eq.~\ref{d2}
This argument is made in the context of degenerate
perturbation theory and the assumptions stated above. 
We now abandon any assumption of perturbation theory and 
simply assume to start with a Hamiltonian which is
the sum of the above two Hamiltonians, i.e.
\begin{eqnarray}
H &=& -t \sum_{<ij>\sigma} c_{i\sigma}^{\dagger}c_{j\sigma}
+ U  \sum_{i} n_{i~\uparrow}  n_{i~\downarrow}\nonumber\\
&&+ J \sum_{<ij>\sigma^{'}} c_{i\sigma^{'}}^{\dagger}c_{j\sigma^{'}}
\sum_{\sigma} c_{i\sigma}^{\dagger}c_{j\sigma}, \nonumber\\
J &=& - g^{2}/U,
\label{d3}
\end{eqnarray} 
where $g$ is some suitable coupling.

	In conclusion we have indicated new ways of
looking at the physics of HTSC and related materials
and in general strongly correlated systems. Although
in its current form quantum groups are restricted to
1-d, we don't see this as a problem but a solution.
This 1-d restriction led us to think that we can
solve a 2-d or 3-d problem by using string theory 
as is done in case of gravity. Gravity is a 4-d 
phenomenon, but can be understood elegantly in 
terms of string theory [1-d objects]. This 
approach is different when one tries to go
from 1-d to higher dimensions directly. Recent
developments in string and topological
field theories can further help us to understand the 
physics of strongly correlated systems. 



 
%\end{itemize} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{enumerate}
%\item{}
%\end{enumerate} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%In a metallic system close to a metal-insulator transition
%the electronic properties play an important and a crucial role.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
\section*{Acknowledgments}
The Sher Alam's work is supported by the Japan Society for
for Technology [JST].
%\begin{thebibliography}{99}
\begin{references}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibitem{alam98}Sher~Alam, {\em A Conjecture for possible theory for the 
description of high temperature superconductivity and 
antiferromagnetism}. Proceedings of Quantum Phenomena in Advanced 
Materials at High Magnetic Fields, 4th International Symposium on Advanced
Physical Fields [APF-4]. KEK-TH-607, KEK Preprint 98-xxx, cond-mat/9812060.
\bibitem{alam99-1}Sher~Alam, {\em Quantum Group based Modelling for the 
description of high temperature superconductivity and
antiferromagnetism}, Phys.~Lett.~{A272},~(2000),~107-112.
~KEK-TH-613, KEK Preprint 98-xxx, cond-mat/9903038.
\bibitem{alam99-2}Sher~Alam et al., ~{\em Theoretical modeling 
for quantum liquids from 1d to 2d dimensional crossover 
using quantum groups}.
KEK-TH-619. KEK Preprint 99-xxx, cond-mat/990345.
\bibitem{fra98} Holger.~Frahm et al., Phys.~Rev.~Lett.
{\bf 81},~(1998),~2116.
\bibitem{alam00-1}Sher~Alam et al., {\em The Choice of
symmetry group for cuprates}, cond-mat/0004269.
\bibitem{alam00-2}Sher~Alam et al., {\em The patching
of critical points using quantum groups}, cond-mat/0004350.
\bibitem{alam00-3}Sher~Alam, {\em Quantum group
conjecture and Stripes} cond-mat/0005168.
\bibitem{alam00-4}Sher~Alam et al., {\em SET based
experiments for HTSC materials} cond-mat/0011037.
\bibitem{kak91}M.~Kaku, {\em Strings, Conformal Fields, and
Topology: An Introduction}, Springer-Verlag, 1991.
\bibitem{yam99}H.~Yamaguchi et al., 
Physica.~{C320},~(1999),~167-172.
%\bibitem{fra98} Holger.~Frahm et al., Phys.~Rev.~Lett.
%{\bf 81},~(1998),~2116.
%\bibitem{kak91}M.~Kaku, {\em Strings, Conformal Fields, and
%Topology: An Introduction}, Springer-Verlag, 1991.
%\bibitem{alam00}Sher~Alam et al., cond-mat/0004269.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\bibitem{fra91}E.~Fradkin, Field Theories of Condensed
%Matter Systems, Addison-Wesley Publishing Company, 1991.
%\bibitem{and87}P.~W.~Anderson, Science {\bf 235},~(1987),~1196.
%P.~W.~Anderson et al., Phys.~Rev.~Lett. {\bf 58},~(1987),~2790.
%\bibitem{hir89} J.~Hirsch and S.~Tang, Phys.~Rev.~Lett.
%{\bf 62},~(1989),~591.
%\bibitem{bal90}A.~P.~Balachandran et al., Hubbard Model
%and Anyon Superconductivity, World Scientific, 1990.
%\bibitem{mor90-91}A.~Moreo et al., Phys.~Rev.~{\bf B41},~(1990),~2313.
%A.~Moreo and D.~J.~Scalapino, Phys.~Rev.~{\bf B43},~(1991),~8211.
%%%%%%%%%%%Experimentst Stripes%%%%%%%%%%%%%%%%%
%\bibitem{oyn99}H.~Oyanagi et al., `Local Lattice Distortions
%in YBa$_{2}$Cu$_{3}$O$_{\rm y}$: Doping dependence, in International
%Conference on Stripes in High T$_{\rm c}$ Superconductivity,
%Rome, June, 1998. Proceedings of STRIPES98, to appear in 
%J.~Superconductivity,~(1999).
%\bibitem{ega94}T.~Egami and S.~J.~L.~Billinge, Prog.~Mater.~Sci
%{\bf 38},~(1994),~359.
%\bibitem{lat92}{\em Lattice Effects in High T$_{\rm c}$
%Superconductors}, ed by Y.~Bar-Yam et al., World Scientific 
%Publishing, Singapore, 1992.
%\bibitem{phs93}{\em Phase Separation in Cuprate Superconductors}, 
%ed by K.~A.~Muller and G.~Benedek, World Scientific 
%Publishing, Singapore, 1993.
%\bibitem{phs94}{\em Phase Separation in Cuprate Superconductors}, 
%ed by E.~Sigmund and K.~A.~Muller, World Scientific 
%Publishing, Singapore, 1994.
%Fukayama one of authors
%\bibitem{fuk92}T.~Tanamoto et al., J.~Phys.~Soc.~Jpn.~{\bf 61},
%\bibitem{fuk93}T.~Tanamoto et al., J.~Phys.~Soc.~Jpn.~{\bf 62},
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\bibitem{zha97}Shou-Cheng Zhang, Science {\bf 275},~(1997),~1089.
%\bibitem{bas97}G.~Baskaran and P.~W.~Anderson, 
%cond-mat/9706076.
%\bibitem{sac95}S.~Sachdev et al., Phys.~Rev.~{\bf B51},~(1995),~14874.
%\bibitem{kak91}M.~Kaku, {\em Strings, Conformal Fields, and
%Topology: An Introduction}, Springer-Verlag, 1991.
%\bibitem{kli97}A.~Klimyk and K.~Schm\"{u}dgen, {\em Quantum Groups
%and their Representations}, Springer-Verlag, 1997.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\bibitem{bas97-1} P.~W.~Anderson and G.~Baskaran, 
%cond-mat/9711197.
%%%%%%%%Section on SO_q(3)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\bibitem{cha89}S.~Chakravarty et al., Phys.~Rev.~{\bf B39},~(1994),~2344. 
%\bibitem{pin94}D.~Pines, Physica {\bf C235},~(1994),~113.
%\bibitem{pin95}A.~V.~Barzykin and 
%D.~Pines, Phys.~Rev.~{\bf B52},~(1995),~13585.
%\bibitem{chu94}A.~V.~Chubukov et al., Phys.~Rev.~{\bf B49},~(1994),~11919. 
%\bibitem{and97}P.~W.~Anderson, The Theory of Superconductivity
%in the High-T$_{c}$ Cuprates.
%(Princeton University Press, Princeton, New Jersey 1999).
%\bibitem{ala98}S.~Alam, {\em A Conjecture for possible theory for the 
%description of high temperature superconductivity and 
%antiferromagnetism}. Proceedings of Quantum Phenomena in Advanced 
%Materials at High Magnetic Fields, 4th International Symposium on Advanced
%Physical Fields [APF-4]. KEK-TH-607, KEK Preprint 98-xxx, cond-mat/9812060.
%\bibitem{don90}S.~Doniach and M.~Inui, Phys.~Rev.~{\bf B41},~(1990),~6668.
%\bibitem{eme95}V.~J.~Emery and S.~Kivelson, Nature~{\bf 374},~(1995),~434.
%\bibitem{zha98}Shou-Cheng Zhang, cond-mat/9808309.
%\bibitem{mos98}A.~S.~Moskvin and A.~S.~Ovchinnikov, 
%Physica {\bf C296},~(1998), 250.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{references}
%\end{thebibliography}
\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\bibitem{Louisell}W.H. Louisell, Radiation and Noise in Quantum Electronics
%(Krieger, New York, 1977) pp. 266-272.
%\bibitem{jay63} E. T. Jaynes and F. W. Cummings, Proc. IEEE 51 (1963) 89.
%\bibitem{mey90} P. Meystre, M. Sargent III, Elements of Quantum Optics.
%(Springer-Verlag, Berlin Heidelberg 1990).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%











cond-mat0102097.extracted_bib
\begin{thebibliography}{99}
\begin{references}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibitem{alam98}Sher~Alam, {\em A Conjecture for possible theory for the 
description of high temperature superconductivity and 
antiferromagnetism}. Proceedings of Quantum Phenomena in Advanced 
Materials at High Magnetic Fields, 4th International Symposium on Advanced
Physical Fields [APF-4]. KEK-TH-607, KEK Preprint 98-xxx, cond-mat/9812060.
\bibitem{alam99-1}Sher~Alam, {\em Quantum Group based Modelling for the 
description of high temperature superconductivity and
antiferromagnetism}, Phys.~Lett.~{A272},~(2000),~107-112.
~KEK-TH-613, KEK Preprint 98-xxx, cond-mat/9903038.
\bibitem{alam99-2}Sher~Alam et al., ~{\em Theoretical modeling 
for quantum liquids from 1d to 2d dimensional crossover 
using quantum groups}.
KEK-TH-619. KEK Preprint 99-xxx, cond-mat/990345.
\bibitem{fra98} Holger.~Frahm et al., Phys.~Rev.~Lett.
{\bf 81},~(1998),~2116.
\bibitem{alam00-1}Sher~Alam et al., {\em The Choice of
symmetry group for cuprates}, cond-mat/0004269.
\bibitem{alam00-2}Sher~Alam et al., {\em The patching
of critical points using quantum groups}, cond-mat/0004350.
\bibitem{alam00-3}Sher~Alam, {\em Quantum group
conjecture and Stripes} cond-mat/0005168.
\bibitem{alam00-4}Sher~Alam et al., {\em SET based
experiments for HTSC materials} cond-mat/0011037.
\bibitem{kak91}M.~Kaku, {\em Strings, Conformal Fields, and
Topology: An Introduction}, Springer-Verlag, 1991.
\bibitem{yam99}H.~Yamaguchi et al., 
Physica.~{C320},~(1999),~167-172.
%\bibitem{fra98} Holger.~Frahm et al., Phys.~Rev.~Lett.
%{\bf 81},~(1998),~2116.
%\bibitem{kak91}M.~Kaku, {\em Strings, Conformal Fields, and
%Topology: An Introduction}, Springer-Verlag, 1991.
%\bibitem{alam00}Sher~Alam et al., cond-mat/0004269.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\bibitem{fra91}E.~Fradkin, Field Theories of Condensed
%Matter Systems, Addison-Wesley Publishing Company, 1991.
%\bibitem{and87}P.~W.~Anderson, Science {\bf 235},~(1987),~1196.
%P.~W.~Anderson et al., Phys.~Rev.~Lett. {\bf 58},~(1987),~2790.
%\bibitem{hir89} J.~Hirsch and S.~Tang, Phys.~Rev.~Lett.
%{\bf 62},~(1989),~591.
%\bibitem{bal90}A.~P.~Balachandran et al., Hubbard Model
%and Anyon Superconductivity, World Scientific, 1990.
%\bibitem{mor90-91}A.~Moreo et al., Phys.~Rev.~{\bf B41},~(1990),~2313.
%A.~Moreo and D.~J.~Scalapino, Phys.~Rev.~{\bf B43},~(1991),~8211.
%%%%%%%%%%%Experimentst Stripes%%%%%%%%%%%%%%%%%
%\bibitem{oyn99}H.~Oyanagi et al., `Local Lattice Distortions
%in YBa$_{2}$Cu$_{3}$O$_{\rm y}$: Doping dependence, in International
%Conference on Stripes in High T$_{\rm c}$ Superconductivity,
%Rome, June, 1998. Proceedings of STRIPES98, to appear in 
%J.~Superconductivity,~(1999).
%\bibitem{ega94}T.~Egami and S.~J.~L.~Billinge, Prog.~Mater.~Sci
%{\bf 38},~(1994),~359.
%\bibitem{lat92}{\em Lattice Effects in High T$_{\rm c}$
%Superconductors}, ed by Y.~Bar-Yam et al., World Scientific 
%Publishing, Singapore, 1992.
%\bibitem{phs93}{\em Phase Separation in Cuprate Superconductors}, 
%ed by K.~A.~Muller and G.~Benedek, World Scientific 
%Publishing, Singapore, 1993.
%\bibitem{phs94}{\em Phase Separation in Cuprate Superconductors}, 
%ed by E.~Sigmund and K.~A.~Muller, World Scientific 
%Publishing, Singapore, 1994.
%Fukayama one of authors
%\bibitem{fuk92}T.~Tanamoto et al., J.~Phys.~Soc.~Jpn.~{\bf 61},
%\bibitem{fuk93}T.~Tanamoto et al., J.~Phys.~Soc.~Jpn.~{\bf 62},
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\bibitem{zha97}Shou-Cheng Zhang, Science {\bf 275},~(1997),~1089.
%\bibitem{bas97}G.~Baskaran and P.~W.~Anderson, 
%cond-mat/9706076.
%\bibitem{sac95}S.~Sachdev et al., Phys.~Rev.~{\bf B51},~(1995),~14874.
%\bibitem{kak91}M.~Kaku, {\em Strings, Conformal Fields, and
%Topology: An Introduction}, Springer-Verlag, 1991.
%\bibitem{kli97}A.~Klimyk and K.~Schm\"{u}dgen, {\em Quantum Groups
%and their Representations}, Springer-Verlag, 1997.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\bibitem{bas97-1} P.~W.~Anderson and G.~Baskaran, 
%cond-mat/9711197.
%%%%%%%%Section on SO_q(3)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\bibitem{cha89}S.~Chakravarty et al., Phys.~Rev.~{\bf B39},~(1994),~2344. 
%\bibitem{pin94}D.~Pines, Physica {\bf C235},~(1994),~113.
%\bibitem{pin95}A.~V.~Barzykin and 
%D.~Pines, Phys.~Rev.~{\bf B52},~(1995),~13585.
%\bibitem{chu94}A.~V.~Chubukov et al., Phys.~Rev.~{\bf B49},~(1994),~11919. 
%\bibitem{and97}P.~W.~Anderson, The Theory of Superconductivity
%in the High-T$_{c}$ Cuprates.
%(Princeton University Press, Princeton, New Jersey 1999).
%\bibitem{ala98}S.~Alam, {\em A Conjecture for possible theory for the 
%description of high temperature superconductivity and 
%antiferromagnetism}. Proceedings of Quantum Phenomena in Advanced 
%Materials at High Magnetic Fields, 4th International Symposium on Advanced
%Physical Fields [APF-4]. KEK-TH-607, KEK Preprint 98-xxx, cond-mat/9812060.
%\bibitem{don90}S.~Doniach and M.~Inui, Phys.~Rev.~{\bf B41},~(1990),~6668.
%\bibitem{eme95}V.~J.~Emery and S.~Kivelson, Nature~{\bf 374},~(1995),~434.
%\bibitem{zha98}Shou-Cheng Zhang, cond-mat/9808309.
%\bibitem{mos98}A.~S.~Moskvin and A.~S.~Ovchinnikov, 
%Physica {\bf C296},~(1998), 250.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{references}
%\end{thebibliography}