%started 17.1.09
\documentclass[11pt,A4]{article}
\usepackage{epsf}
\usepackage{graphicx} % standard LaTeX graphics tool
%\input epsf
\textheight = 24truecm
\textwidth = 16.2truecm
\hoffset = -1.8truecm
\voffset = -2truecm
\def\ll{\label}
\def\re{\ref}
\def\c{\cite}
\def\b{\begin}
\def\r1{(\ref{$1})}
\def\ot{\otimes}
\def\nn{\nonumber}
\def\sn{\rm sn}
\def\pa{\partial}
\def\kap{\kappa}
\def\ms{\medskip}
\def\cR{\cal R}
\def\cF{\cal F}
\def\cP{\cal P}
\def\ti{\tilde}
\def\cn{\rm cn}
\def\dn{\rm dn}
\def\ga{\gamma}
\def\ep{\epsilon}
\def\th{\theta}
\def\ba{\begin{array}{c}}
\def\e{\end}
\def\sk{\smallskip}
\def\ea{\end{array}}
\def\pr{\prod}
\def\ni{\noindent}
\def\si{\sigma}
\def\da{\dagger}
\def\De{\Delta}
\def\de{\delta}
\def\bet{\beta}
\def\ov{\over}
\def\ha{{1\over 2}}
\def\qr{{1\over 4}}
\def\l{\left}
\def\l({\left(}
\def\r){\right)}
\def\r{\right}
\def\rw{\rightarrow}
\def\om{\omega}
\def\la{\lambda}
\def\al{\alpha}
\def\sec{\section}
\def\be{\begin{equation}}
\def\bc{\begin{center}}
\def\ec{\end{center}}
\def\bit{\begin{itemize}}
\def\eit{\end{itemize}}
\def\ee{\end{equation}}
\def\ed{\end{document}}
\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}
\def\efr{\end{flushright}}
\def\nn{\nonumber\\}
%======================== journal macros ===============================
\begin{document}
\thispagestyle{empty}
\begin{center}
{\footnotesize Available at:
{\tt http://publications.ictp.it}}\hfill IC/2010/023\\
\vspace{1cm}
United Nations Educational, Scientific and Cultural Organization\\
and\\
International Atomic Energy Agency\\
\medskip
THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS\\
\vspace{2.5cm}
{\bf NONULTRALOCAL QUANTUM ALGEBRA AND 1D ANYONIC\\
QUANTUM INTEGRABLE MODELS}\\
\vspace{2cm}
Anjan Kundu\footnote{Senior Associate of ICTP. anjan.kundu@saha.ac.in}\\
{\it Theory Group \& CAMCS,
Saha Institute of Nuclear Physics,\\
Calcutta, India\\
and\\
The Abdus Salam International Centre for Theoretical Physics,
Trieste, Italy.}
\end{center}
\vspace{1cm}
\centerline{\bf Abstract}
\baselineskip=18pt
\bigskip
Applying braided Yang-Baxter equation
quantum integrable and Bethe ansatz, solvable 1D anyon lattice and field
models are constructed.
Along with known models we discover novel lattice anyon and q-anyon models as well as nonlinear Sch\"odinger equation
(NLS) and the derivative NLS anyon quantum field models,
N-particle sectors of which yield the well-known anyon gases,
interacting through $\delta $ and derivative $ \delta$ function
potentials.
As a byproduct we discover a new anyon quantum group Hopf algebra with
unusual braided multiplication.
\vfill
\begin{center}
MIRAMARE -- TRIESTE\\
May 2010\\
\end{center}
\vfill
%{PACS:} 02.30.lk,
%integrable system
%05.30.Pr
%anyon
%02.30.jr,
%PDE
% 05.45.Yv,
%soliton
%11.10.Lm,
%nonlin nonloc FT
%{\it Key words}:\\
% Nonultralocal model, Braided YBE, Quantum integrability, 1D anyon
%and q-anyon
%lattice models, Anyonic NLS and derivative NLS field models, algebraic Bethe
%ansatz, Anyon quantum group
\newpage
\section{Introduction}
\label{sec1}
Anyons \cite{wilcz82} are receiving renewed attention
after their experimental confirmation \cite{anyExp} and the promise of potential
applications in quantum computation \cite{anyKitaev}.
Although the anyons live
in two
space-dimensions, they remarkably retain their basic properties
when projected to one-dimension (1D)
\cite{korany}. Therefore, since
exactly solvable models can be constructed in 1D, interacting
1D anyon models introduced in \cite{kun99,kunbach08,anyspin08}
are becoming increasingly popular
in recent years
\cite{korany,1Dany}.
However, though these models
capture the exchange algebra of anyon operators at the space-separated
points $x \neq y $, they cannot reproduce the required
anyonic commutation relation (CR) at $x = y $ and hence
they behave like bosons
\cite{kun99,kunbach08,korany,1Dany} or fermions \cite{anyspin08} at the coinciding
points and consequently cannot
interpolate between bosons and fermions in
the entire domain. This remains as a major drawback of the existing 1D anyon
models.
Another unsettled problem is that, unlike the
nonlinear Sch\"odinger equation (NLS) and the derivative
NLS, which give the known solvable Bose gases
\cite{LiebLin65,Snirman94} at their N-particle sectors, no
anyon quantum field models are discovered yet, which could yield
the known anyon gases at the N-particle case,
\cite{kun99,kunbach08}.
Our aim here is to resolve both these unsolved problems by finding
quantum integrable novel anyonic lattice and field models, based on a
braided extension of the Yang-Baxter equation (BYBE)
\cite{bybe,nulmkdv}. Note that
the anyons,
not commuting at space-separated points,
belong to nonultralocal models and go beyond the standard
formulation of the quantum inverse scattering method \cite{Fadrev}.
Remarkably, our approach through the BYBE
leads also to the discovery of a new anyon
quantum group exhibiting Hopf algebra properties with
nonultralocal braiding
relations.
The arrangement of the paper is as follows. Sect. 2 describes the
brief history of 1D anyon models.
Sect. 3 describes the BYBE and the reletad quantum integrability.
Subsections 3.4-5 construct the quantum integrable lattice
anyon and the NLS anyon field model. Subsections 3.6-8 account for the
q-anyon and the derivative NLS anyon field model. Sect. 4
presents the anyon quantum group with Hopf algebra structure.
Sect. 5 is the concluding section
followed by the bibliography.
\section{Exactly solvable 1D boson and anyon gases}
Surprisingly, anyons continue to
exhibit nontrivial exchange and cross-over properties even
when projected to 1D, with
the 2-particle anyon wave function showing
\be \Phi (x_{1},x_{2}) =e^{-i \theta}
\Phi (x_{2},x_{1})
,\ll{any1} \ee
under exchange and an intriguing sensitivity on the boundary condition
on a chain of length $L $~\cite{korany}
\be
\Phi (x_{1}+L,x_{2}) = e^{-2i \theta}
\ \Phi (x_{1},x_{2}+L), \ll{any2}\ee
confirming that the {\it
passing } of particle 1 through 2 in 1D is not the same as 2 {\it
passing } through 1. This reflects the known property of the standard 2D anyon,
where the effect of particle 1 going around 2 is different from
that of 2 going round 1. Note that from (\re{any1} -\re{any2})
one recovers the usual bosonic behavior at
$ \theta=0, $, while $\theta=\pi $
corresponds to the fermions.
The focus on 1D anyons has been intensified during the recent years, since
together
with the preserving of the basic properties of the standard anyons, they could be
exactly solvable in 1D, offering detailed analytic result, which
should be valuable for analysing standard anyons.
Apart from the well-known Calogero model with anyon-type exchange
statistics,
there is an interesting history of 1D
anyon models belonging to the exactly solvable class.
\subsection{Exactly solvable Bose gases}
Bose gas interacting through $\de $-function potential is a celebrated
exactly solvable model introduced by Lieb \& Liniger \cite{LiebLin65}
\be {H}^{(b1)}_N
=-\sum_k^N \partial^2_{x_k}+ \sum_{}
c \de ({x_k-x_l}).\ll{boseD}\ee
After about thirty years another
solvable Bose gas model, interacting through
derivative $\de $-function potential
was also proposed \cite{Snirman94}
\be {H}^{(b2)}_N
=-\sum_k^N \partial^2_{x_k}+ \sum_{}
i \kap \de ({x_k-x_l})\left(
(\partial_{x_k}+\partial_{x_l})\right)
\ll{bosedD} \ee
\subsection{ Exactly
solvable anyon gases and $N$-particle anyon models}
After the success of Bose gases with singular potentials, there were
naturally attempts
to build Bose gases with even higher singular potentials,
like {\it double} $\de $-function potentials
\[
\gamma_1 \sum_{}
\de ({x_j-x_k})
\de ({x_l-x_k}) + \gamma_2 \sum_{}
(\de ({x_k-x_l}))^2,
\]
which, however, remained unsuccessful until
the introduction of a $\de $-function anyon gas model \cite{kun99}.
This exactly solvable 1D anyon model was
shown, in fact, to be
equivalent to a double $\de $-Bose gas with
its coupling constants related by $ \gamma_1=\gamma_2=\kappa^2 $.
Recently, another exactly solvable 1D anyon gas model interacting through
derivative-$\de $-function potential is also proposed \cite{kunbach08}.
These anyon gas models, with
different research groups focusing on their various aspects, are
gaining growing popularity \cite{1Dany}.
In a recent work \cite{anyspin08}
other nearest neighbor lattice anyon models, solvable by the algebraic Bethe ansatz, were proposed.
However, as mentioned, all 1D anyon models have an inherent drawback: they
behave like bosons or fermions at the coinciding points.
It is well known that there are quantum integrable field models,
$N $-particle sectors
of which correspond to the exactly solvable Bose gases.
In fact the nonlinear Sch\"odinger equation (NLS)
\be {H}^{(b1f)}
=\int dx ( \psi^\dag_x \psi_x + c (\psi^\dag \psi)^2)
\ll{bNLS} \ee
in bosonic field
$ [\psi (x), \psi^\dag(y)]= \de(x-y)$ corresponds to
the $\de $-Bose gas, while the $N$-particle model of the derivative NLS to
the derivative $\de $-function Bose gas.
However, an important question surrounding the 1D anyon gases,
that remained unanswered up to this date
is
that, what are
the anyonic QFT models, $N $-particle sectors of which
would generate the anyon gases interacting through $\de $ and
derivative $\de $ -function potentials. Our aim here is to
take up this challenging problem and
discover the needed
quantum integrable anyonic QFT models.
We also intend to
construct integrable anyonic lattice and field models
in a systematic way through Yang-Baxter equation, so as to guarantee
anyonic commutation relations at all points, including the coinciding
points, thus resolving the existing problem of the anyon models, not
obeying anyon exchange at the coinciding points.
\section {Construction of integrable anyon models through braided
Yang-Baxter equation }
Anyon models due to the noncommutation of anyon fields at space
seperated points belong to the problem of nonultralocal models and
go beyond the
standard formulation of quantum integrable systems based on the YBE.
We have to
use therefore an extension of the YBE with nontrivial braiding
(BYBE) developed by us \cite{bybe}, for
systematic generation of the anyon commutation relations as well as for the construction of
quantum integrable anyon models.
\subsection{Braided Yang-Baxter equation}
The BYBE represents two different commutation relations for the Lax operator $
L_{aj}(u)$,
given at the coinciding and at noncoinciding points, expressed through the
standard quantum ${R}(u-v) $-matrix in addition to a braiding matrix $Z $:
\be
{R}_{12}(u-v)Z_{21}^{-1}(L_{1j}(u) Z_{21} L_{2j}(v)
= Z_{12}^{-1}L_{2j}(v) Z_{12} L_{1j}(u){R}_{12}(u-v), \ll{bybe} \ee
at the lattice sites $\ \ j=1, 2, \ldots. N $,
together with the
braiding relation (BR):
\be
L_{2 k}(v) Z_{21}^{-1}L_{1 j}(u)
= Z_{21}^{-1}L_{1 j}(u)Z_{21} L_{2 k}(v) Z_{21}^{-1}
\ll{br}\ee
for $k>j$, representing nonultralocality, i.e. noncommutativity
at space separated points.
Recall that the quantum $R(u-v) $ matrix is a $4 \times 4 $ matrix
\begin{equation}
\label{R}
R(\lambda)=\left(
\begin{array}{llllll}
a(\lambda) \ \quad \ \quad \ \quad \\
\quad \ b(\lambda) \ \ c \ \ \quad \\
\quad \ \ c \ \ b(\lambda) \ \quad \\
\quad \ \ \quad \ \ \quad \ \ a(\lambda)
\end{array}\right)
\end{equation}
with rational: \be a(\lambda)=\lambda+\alpha, \ b(\lambda)= \lambda, \ c=
\alpha \ll{Rrat} \ee or trigonometric: \be a(\lambda)= \sin (\lambda+\alpha), \ b(\lambda)=
\sin \lambda, \ c=
\sin \alpha \ll{Rtrig} \ee solutions.
We consider both of these forms and show that
they would generate two different classes of anyonic integrable models.
The braiding matrix $Z$ containing the anyonic parameter $\theta $, may be
given in the graded form
\[ Z=\sum_{a,b} e^{i\theta (\hat a \cdot \hat
b)}e_{a,b}\otimes e_{b,a} , \ \ \hat a=0,1 \ \mbox{denotes
gradings}, \]
which satisfies all the relations as required for the braided
generalization \cite{bybe}.
For a $4 \times 4 $ matrix with the choice $\hat 1 =0, \ \hat 2=1 $
we get the simplest form
\be Z=diag (1,1,1, e^{i \theta}) \ll{Z}\ee
which we use in constructing all our anyonic models. It is evident that
for $ \theta=0: Z=I, \ $ BYBE (\re {bybe}) reduces to the
standard YBE
$ \
{R}(u-v)L_{1j}(u) L_{2j}(v)
= L_{2j}(v) L_{1j}(u){R}(u-v), \ $
while the BR (\re {bybe}) recovers the
ultralocality condition
$[ L_{2 k}(v),L_{1 j}(u)]=0, \ k \neq j $, related to the
bosonic commutativity.
\subsection {Quantum integrable model construction}
For building the Hamiltonian of the model we have to
construct conserved quantities by switching over from the local to a
global picture, by defining the transfer matrix as a global quantum operator
acting on the multi-particle Hilbert space:
\[ \tau(u) = trace_a({L}_{a1}(u)\ldots {L}_{aN}(u)) ,\]
which generates all conserved quantities
$ \ \log \tau(u)=\sum_n C_n u^n $.
The BYBE guarantees that $[\tau(u),\tau(v)]=0 $, and hence
$[C_n,C_m]=0 $, ensuring the
quantum
integrability of the model, while the Hamiltonian of the model can be chosen
as any of the conserved operators: $H=C_n , \ \ n=1, 2, 3, \ldots $.
The Lax operator $L_{j}(u) $ may be constructed as a solution of the BYBE
(\re {bybe}) using the known solution of the
quantum $R $-matrix and the braiding matrix $Z $.
Let us consider first the rational $ R$-matrix solution (\re {R}-\re{Rrat})
together with the
$Z$-matrix (\re{Z}).
The anyonic commutation relations (CR) are obtained directly from the BYBE and
the BR, different realizations of
which construct different types of anyon algebra.
\subsection{ Lattice hard-core anyon model }
We can construct through the above scheme a
nearest-neighbor interacting anyon model, proposed recently
\cite{anyspin08} as
\be C_1= H^{(1a)}
= \sum _{k=1}^N 2 n_kn_{k+1}+
+a_ka^\dagger_{k+1}+a^\dagger_k a_{k+1}
, \ \ n_k \equiv a^\dagger_k a_{k},\ll{NNany}\ee
with anyonic CR at space-separated points $k>l $:
\be a_k a_l^\dagger=e^{i \theta} a_l^\dagger a_k, \ \ a_k a_l=e^{-i \theta} a_l a_k
\ll{anycrHC}\ee
However, at the coinciding points one
gets only fermionic relations
$ [a_k , a_k^\dagger]_+=1,
\ \ a^2=0, $
confirming the drawback as mentioned above.
\subsection{ Novel anyon lattice model }
As a different realization of the BYBE we construct another quantum
integrable anyon model
with next-nearest neighbor and higher order nonlinear
interactions given by the Hamiltonian
\be C_3= {\rm H}^{(2a)}= \sum_k (\psi^\dagger_{k+1}
\psi_{k-1} \\ -(n_{k}+
n_{k+1})\psi^\dagger_{k+1}
\psi_{k} + \frac 1{3 \Delta ^2} n^3_k, )
\ll{anyLNLS} \ee with $n_k=p_{k}+ \Delta ^2
\psi^\dagger_{k}
\psi_{k}$, with $p_{k} $ related to the number operator.
It is remarkable that this model gives finally
the needed anyonic CR at the coinciding points $k$:
\be
\psi_{k} \psi_{k} ^{\dagger}-
e^{-i \theta}\psi_{k} ^{\dagger} \psi_{k} = p_{k} \frac 1 \Delta
\ll{anyCRx0} \ee
together with at noncoinciding points $k>j $: \be
\psi_{k} \psi ^{\dagger }_{j}=e^{i \theta}
\psi ^{\dagger}_{j} \psi_{k}. \ll{anyCRx+} \ee
Thus we solve one of the existing problems of
the anyon models by achieving
the anyonic CR at all points.
\subsection{Quantum integrable NLS anyon field model}
Taking carefully the continuum limit of the above
lattice anyon model (\re{anyLNLS}) with the
lattice constant $\Delta \to 0
$, $ k \to x, $ and the field $
\psi_k \to A(x), $ we can derive a
novel NLS anyon field model
\be
\hat H^{(3a)}= \int dx (A^\dagger_{x}A_x+c (A^\dagger A)^2 ) \ll{anyLNS}
\ee
with the anyon field operator $A(x) $ obeying again
the needed CR at all points, obtained from its lattice counterpart
(\re{anyCRx0},\re{anyCRx+}) at the coinciding points $x\to y $
\be
A(x) A^\dagger(y)-e^{i \theta}A^\dagger(y)A(x)=\delta(x-y) \ll{anyx0} \ee
and at $\ x>y $:
\bea A(x) A^\dagger(y)=e^{i \theta} A^\dagger(y) A(x), \ll{anyx1+} \\
A(x) A(y)=e^{-i \theta} A(y) A(x), \ll{anyx2+} \eea
Clearly these anyonic CR can interpolate between
bosons (at $\theta=0$) and fermions (at $\theta=\pi$)
at all points.
Now we solve another outstanding problem mentioned in the introduction
by finding the
$N$-particle sector
\be
|N>=\int d^N x \sum_{\{x_l\}} \Phi (x_1,x_2,\ldots x_N)
A^\dagger (x_1)A^\dagger (x_2)\cdots A^\dagger (x_N) |0> \ll{Nsector}\ee
of the NLS anyon field model (\re{anyLNS}), which gives indeed
the well-known $\de $ -function anyon gas model
\[ H_N= -\sum_k \partial^2_k +c \sum_{k \neq j} \delta (x_k -x_l)
\]
establishing thus its missing link to a genuine 1D anyon quantum field model.
All the above anyon models are obtained using the rational $R$-matrix
solution. Now we switch over to the trigonometric case.
\subsection{Trigonometric class of anyon models}
Considering the trigonometric quantum $R$-matrix (\re{R},\re{Rtrig})
related to the $ xxz$ spin-$\frac 1 2 $ chain, but keeping the same braiding
matrix $Z $ (\re{Z}), we generate here a new trigonometric
class of anyon lattice and field
models, with a deformation parameter $q=e^{i \alpha} $.
\subsection{q-anyon model}
Following the same construction rule but using
now $R$-matrix (\re{Rtrig}), we get from BYBE (\re{bybe}) a
new
{\it anyonic q-oscillator} model with CR:
\be \phi_{k} \phi_{k} ^{\dagger}-
e^{i \theta} \ \phi_{k} ^{\dagger} \phi_{k} = e^{i \theta N_k}
\cos 2 \alpha N_k \ll{qanyx0} \ee
and
\be \phi_{k} \phi_{j} ^{\dagger}=
e^{i \theta }
\phi_{j} ^{\dagger}\phi_{k}, \ll{qanyx+} \ee
at the space-separated points $ k >j$. We do not present here further details of this
anyon q-oscillator model, however going to its continuum limit we derive another new
anyonic quantum field model.
\subsection{Derivative NLS anyon fields model}
At the field limit $\De \to 0 , \ \phi_k \to D(x), $
we obtain from the anyon q-oscillator a novel
quantum integrable derivative- NLS anyon field model
\be \hat H^{(4a)}= \int dx
(D^\dagger_{x}D_x+2i \kappa (D^\dagger )^2 D D_x) \ll{anyDNLS} \ee
with the anyon field operator satisfying the CR
\be
D(x) D^\dagger(y)-e^{i \theta}D^\dagger(y)D(x)=\kappa \delta(x-y)
\ll{anyCRx0dnls} \ee
and
\be D(x) D^\dagger(y)=e^{i \theta} D^\dagger(y) D(x),
\ll{anyCRx+dnls} \ee
at $\ x > y $.
The $N$-particle sector $|N> $ of this anyon DNLS field model
gives interestingly the
recently proposed derivative- $\de $ function anyon gas model
\[ H^d_N= -\sum_k \partial^2_k +i \kappa
\sum_{k \neq j} \delta (x_k -x_l) (\partial_{x_k} + \partial_{x_l}),
\]
establishing the final missing link of this anyon gas
to a new quantum integrable anyon field model.
Thus we have constructed from the BYBE above,
a series of anyonic and q-anyonic models, namely
i) nearest-neighbor hard-core anyon, ii) next-nearest-neighbor
nonlinear lattice anyon, iii) quantum NLS anyon field model, iv) anyon q-oscillator
and v) anyonic DNLS quantum field model. We emphasize that
all these models built systematically
from the nonultralocal BYBE and the braiding relations
are
quantum integrable models, which are exactly solvable by
the {\it algebraic Bethe ansatz}.
\section { Novel anyonic quantum group with nonultralocal braiding }
In constructing the anyonic q-oscillator models we have used the
trigonometric $R$-matrix (\re{R},\re{Rtrig}) together with the braiding
matrix (\re{Z}) in BYBE (\re{bybe}) and BR (\re{br}).
It is intriguing that, following the same procedure but taking a different realization of the
anyon algebra, we can discover a novel
{\it anyon quantum group} algebra $A_\theta su_q(2) $, with an unusual nonultralocal
braiding relation.
This algebra, deformed by two independent parameters $q=e^{i \alpha}$
and the anyonic parameter
$s=e^{i \theta}$
exibits also a beautiful Hopf algebra structure
with nontrivial coproduct and multiplication.
The two-parameter deformed anyon quantum group may be expressed through the
algebraic relations
\bea S^{+}S^{-} -s S^{+}S^{+}
=[2S^3]_q s^{-S^3 }, \nonumber \\
q^{S^3}S^{\pm} = q^{\pm 1} S^{\pm}q^{S^3}, \ \
s^{S^3}S^{\pm} = s^{\pm 1} S^{\pm}s^{S^3}
\ll{alganyQG}\eea
denoting $ \ [x]_q \equiv \frac {q^{x}-q^{-x}} {q^{}-q^{-1}}=
\frac {\sin \alpha x} {\sin \alpha} $.
Remarkably, this algebra also exhibits the Hopf algebra structure
with an
unusual braided multiplication
\bea (I \otimes S^{\pm})(S^{\pm} \otimes I)=
s^{-1 }(S^{\pm} \otimes S^{\pm}),
\nonumber \\
(I \otimes S^{\mp})(S^{\pm} \otimes I)=
s(S^{\pm} \otimes S^{\mp})
\ll{multanyQG}\eea
and an intriguing two-parameter deformed coproduct structure
\bea \Delta (S^{+})= q^{-S^3} \otimes S^{+}+ S^{+}\otimes q^{S^3} s^{-S^3},
\nonumber \\
\Delta (S^{-})= q^{-S^3} s^{-S^3} \otimes S^{-}+ S^{-}\otimes q^{S^3},
\ \ \Delta (S^3)= S^3 \otimes I+ I\otimes S^3.
\ll{coprodanyQG}\eea
It is interesting to check by direct insertion
that the coproducts
$\Delta(S^\pm ),\Delta (S^3) $ do satisfy the same algebra, where
the factors with $s$-parameter in the algebraic relations
conspire with those in the multiplication and in the coproduct, such that all extra factors cancel out, similar to the
standard case.
At $s=1 $ we clearly recover from (\re {alganyQG}) the standard quantum algebra
\[ [S^-, S^+]_- = [2S^3 ]_q, \ \ \ [S^3,S^\pm]=\pm S^3, \]
at the coinciding points and the trivial commutative multiplication \ \[
(I \otimes S^{\pm})(S^{\pm} \otimes I)=
(S^{\pm} \otimes S^{\pm}),
\ \ \
(I \otimes S^{\mp})(S^{\pm} \otimes I)=
(S^{\pm} \otimes S^{\mp}), \]
at different points $ \ k >j $.
At $q \to 1 $ with arbitrary anyon parameter $ s$
we get on the other hand a pure anyonic-deformed $A_\theta su(2) $ algebra
\bea S^{+}S^{-} -s S^{+}S^{+}
=2S^3 s^{-S^3 }, \nonumber \\
s^{S^3}S^{\pm} = s^{\pm 1} S^{\pm}s^{S^3}
\ll{algany}\eea
again with the same braided multiplication rules (\re{multanyQG})
and a { twisted coproduct}
\bea \Delta (S^{+})= I \otimes S^{+}+ S^{+}\otimes s^{-S^3},
\nonumber \\
\Delta (S^{-})= s^{-S^3} \otimes S^{-}+ S^{-}\otimes I,
\ \ \Delta (S^3)= S^3 \otimes I+ I\otimes S^3
\ll{coprodany}\eea
Note that at $ s=\pm 1$ this anyonic spin algebra would be reduced to
the standard or an anticommuting spin algebra, as evident
from (\re{algany}). The corresponding Hopf algebra structures are also
obtained from (\re{coprodany},\re{multanyQG}).
\section{Concluding remarks}
We have constructed two classes of 1D anyon models, rational and
trigonometric, in a systematic
way starting from the braided YBE. The known, as well as the new anyon models
that
we have constructed, are all quantum integrable
and exactly solvable by the algebraic Bethe Ansatz.
Among the new models a next-nearest-neighbor interacting lattice anyon
model and a nonlinear Schr\"odinger anyon quantum field model belong
to the rational class, with the latter model being the missing link
to the known $\de $ anyon gas. THE q-anyon and the derivative NLS
anyon field model that we discover belong to the triogonometric class,
where the dNLS anyon model provides the needed link to the
recently proposed derivative $\de $- function anyon gas.
Significantly, the
anyons in the new models exhibit proper anyonic CR at all points, rectifying thus
the deficiency of the existing anyon models.
We also obtain from the BYBE in the trigonometric case an importantly
new anyon quantum group Hopf algebra, which generalizes the known quantum
algebra as well as its coalgebra structures
with an additional deforming anyonic parameter, resulting in a novel
nonultralocal multiplication and an unusual twisted coproduct.
It would be highly motivating to investigate various aspects of this Hopf
algebra as well as to find representations of this new two-parameter
algebra, especially at the parameters $q,s $ at the roots of unity.
Another promising line of research would be to find nonabelian realizations
of the integrable 1D anyon models exploiting the BYBE, which might shed
light to the nonabelian anyon models, importance of which is growing in
recent years \cite{anyKitaev}.
\section*{Acknowledgments}
This work was done within the framework of the Associateship Scheme of the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy.
\newpage
\begin{thebibliography}{99}
\bibitem{wilcz82} F. Wilszek, {\it Phys. Rev. Lett.}, {\bf 48} (1982) 1144
\bibitem{anyExp} F. F. Camino, W. Zhou, V. J. Goldman, {\it Phys. Rev.}, {\bf B 72} (2005) 075342
E. Kim, M. Lawer, S. Vishweshwara and E. Fradkin, {\it Phys. Rev.
Lett.}, {\bf 95} (2005) 176402
\bibitem{anyKitaev} A. Kitaev, {\it Ann. Phys.}, {\bf 303} (2003) 2;
{\bf 321} (2006) 2
S. Trebst, M. Troyer, Z. Wang, and A. W. Ludwig.
arXiv:0902.3275 (2009)
C. Nayek, S.U. Simon, A. Stern, M. Freedman and S. Das Sarma, {\it Rev. Mod.
Phys.}, {\bf 80} (2008) 1083
\bibitem{korany} O.I. P\^{a}{t}u, V.E. Korepin and D.V. Averin, {\it J.\
Phys.}, {\bf A 40} (2007) 14963
\bibitem{kun99} Anjan Kundu, {\it Phys. Rev. Lett.}, {\bf 83} (1999) 1275
\bibitem{kunbach08} M. T. Batchelor, X.-W. Guan, A. Kundu, {\it J. Phys.} (FTC),
{\bf A 41} (2008) 352002
\bibitem{anyspin08} M. T. Batchelor, A. Foerster, X-W. Guan, J. Links and H-Q. Zhou,
{\it J. Phys.}, {\bf A 41} (2008) 465201
\bibitem{1Dany}
M. T. Batchelor, X. W. Guan and N. Oelkers, {\it Phys. Rev. Lett.}, {\bf
96} (2006)
210402-1
M. D. Girardeau, {\it Phys. Rev. Lett.}, {\bf 97}, (2006) 100402-1
D.V. Averin and J.A. Nesteroff, {\it Phys.\ Rev.\ Lett.
}, {\bf 99}, (2007) 096801
M.T. Batchelor, X.-W. Guan, and J.-S. He, {\it J.\ Stat.\
Mech.} P03007 (2007)
P. Calabrese and M. Mintchev, {\it Phys.\ Rev.}, {\bf 75}
(2007) 233104
R. Santachiara, F. Stauffer and D.C. Cabra, {\it J.\ Stat.\
Mech.} L06002 (2006)
O.I. P\^{a}{t}u, V.E. Korepin and D.V. Averin, {\it J.\
Phys.}, {\bf A 41} (2008) 145006, 255205
\bibitem{LiebLin65}E. H. Lieb and W. Liniger, {\it Phys. Rev.}, {\bf 130} (1963) 1605
\bibitem{Snirman94} A. G.
Shnirman, B. A. Malomed and E. Ben-Jacob, {\it Phys. Rev.}, {\bf A 50} (1994) 3453
\bibitem{bybe}
L Hlavaty and Anjan Kundu, {\it Int. J. Mod. Phys.},
{\bf A 11} (1996) 2143-2165
\bibitem{nulmkdv}
Anjan Kundu, {\it Mod. Phys. Lett.}, {\bf A 10} (1995) 2955-2966
\bibitem{Fadrev} L. D. Faddeev, {\it Sov. Sc. Rev.}, {\bf C1} (1980) 107
P. Kulish and E. K. Sklyanin,
{\it Lect. Notes in Phys.} (ed. J. Hietarinta et al., Springer, Berlin, 1982)
{\bf 151}
p. 61
\end{thebibliography}
\end{document}