
\documentstyle[epsfig,preprint,aps]{revtex}
\begin{document}
\preprint{CBPF-NF-005/03 or hep-th/0301114}
\tightenlines
\title{$q$-DEFORMED KINK SOLUTIONS}
\author{A. F. de
Lima$^{(a)}$ and R. de Lima Rodrigues$^{(b)}$\thanks{Permanent address:
Departamento de Ci\^encias Exatas e da Natureza, Universidade Federal de
Campina Grande, Cajazeiras, PB -- 58.900-000 -- Brazil.
E-mail for RLR are: rafael@fisica.ufpb.br and rafaelr@cbpf.br.
E-mail for AFL is
aerlima@df.ufpb.br.}\\
{}$^{(b)}$Departamento de F\'\i sica, Universidade Federal de Campina
Grande \\
 Campina Grande, PB -58.109-970 -- Brazil\\
{}$^{(b)}$ Centro Brasileiro de Pesquisas F\'\i sicas, Rua Dr. Xavier
Sigaud, 150\\ Rio de Janeiro-RJ-22290-180, Brazil}

\maketitle

\begin{abstract}
The $q$-deformed kink of the $\lambda\phi^4-$model is obtained via the
normalisable ground state eigenfunction of a fluctuation operator
associated with
the $q$-deformed hyperbolic functions. The kink mass, the bosonic zero-mode
and the $q$-deformed potential in 1+1 dimensions are found.
\end{abstract}

\vspace{1cm}
PACS numbers: 11.30.Pb, 03.65.Fd, 11.10.Ef.




\newpage

\section{introduction}

The kink of a scalar potential in 1+1 dimensions is a static,
non-singular, {\it classically stable} and a finite localized energy
solution of the equation of motion, which can be in topologically stable
sectors \cite{raja}.
In a recent lecture \cite{vacha02},
an investigation on the topological defects starting with the simplest case
of domain walls was presented, and then considerations to more
elaborate and realistic models were put forward.

In the present letter, one works
with the algebraic technique of the SUSY QM  formulated by Witten
\cite{Witten81,La,Fred}, which is associated with a second order
differential equation for the $q$-deformed hyperbolic functions \cite{arai}.
Recently, the $q$-deformed hyperbolic function was used to construct a new
$\eta-$pseudo-Hermitian
 complex potential with PT symmetry \cite{sun02}. Other potentials
like Rosen-Morse well, Scarf, Eckart and the generalized P\"oschl-Teller
were constructed  via shape invariance \cite{jia02}.


The stability equation for topological and non-topological solitons has been
approached in the framework of supersymmetric quantum mechanics (SUSY QM)
\cite{Talu,Kuls,Sukumar86,K87,R95,vvr02}. The marginal stability and the
metamorphosis of Bogomol'nyi-Prasad-Sommerfield (BPS) states
have been investigated, via SUSY QM, and  presented a detailed analysis for
a 2-dimensional $N=2-$Wess-Zumino model with two chiral superfields,
and composite dyons in $N=2$-supersymmetric gauge theories \cite{Shif01b}.

In this letter, the interesting program of proposing a new potential model in
1+1 dimensions, whose essential point is associated with the translational
invariance of the $q$-deformed kink solutions, is investigated.


\section{Solitons in 1+1 dimensions}

Consider the Lagrangian density for a single scalar field, $\phi(x,t),$ in
(1+1)-dimensions, in natural system, given by

\begin{equation}
{\cal L}\left(\phi, \partial_{\mu}\phi\right) = \frac{1}{2}
\partial_{\mu}\phi\partial^{\mu}\phi - V\left(\phi\right)
\label{E1},
\end{equation}
where  $V(\phi)$ is any positive semi-definite function of $\phi$, which
must have at least two zeroes for soliton solution to exist. It represents a
well-behaved potential energy. However, as it will be shown below, we have
found a new potential which is exactly solvable in the context of
the classical theory in (1+1)-dimensions.

The field equation for a static classical configuration, $\phi =
\phi_c\left(x\right),$ becomes

\begin{equation}
\label{E2}
-\frac{d^2}{dx^2} \phi_c\left(x\right) +
\frac{d}{d\phi_c}V\left(\phi_c\right) =0, \qquad \dot\phi_c = 0,
\end{equation}
with the following boundary conditions: $\phi_c(x)\rightarrow
\phi_{vacuum}(x)$ as $x\rightarrow \pm \infty.$



Since the potential is positive, it can be written as

\begin{equation}
\label{E4} V(\phi) = \frac{1}{2} U^2(\phi).
\end{equation}
Thus, the total energy for the $q$-kink becomes

\begin{eqnarray}
\label{ET}
E&&=\frac 12\int\left[\left(\phi^{\prime}\right)^2+U^2\right]dx\nonumber\\
{}&&=\frac 12\int\left[\left(\phi^{\prime}\mp U\right)^2\pm
2U\phi^{\prime}\right]dx.
\end{eqnarray}
In this case, the  Bogomol'nyi form of the energy, consisting of a
sum of squares and surface terms, becomes


\begin{equation}
\label{Ebogo}
E\geq\left|\int dx\frac{\partial}{\partial x}U[\phi(x)]\right|,
\end{equation}
under the well-known Bogomol'nyi condition for the kink solution,

\begin{equation}
\label{E5}
\frac{d\phi}{dx} = \pm U(\phi)
\end{equation}
where the solutions with the plus and minus signs represent two static
field configurations.


\section{Stability Equation}

The classical stability of the soliton solution is investigated by
considering small perturbations around it,

\begin{equation}
\label{E11} \phi(x,t) = \phi_c(x) + \eta (x,t),
\end{equation}
where we expand the fluctuations in terms of the normal modes,
\begin{equation}
\label{E12} \eta (x,t) = \sum_n \epsilon_n \eta_n (x) e^{i\omega_n t},
\end{equation}
with the $\epsilon_n'$s chosen so that $\eta(x,t)$ is real. A localized
classical configuration is said to be dynamically stable if the fluctuation
does not destroy it. The equation of motion becomes a Schr\"odinger-like
equation, viz.,


\begin{equation}
\label{E13}
O_F\eta_n (x) = \omega_n{^2}\eta_n (x), \quad
O_F=-\frac{d^2}{dx^2}+ V^{\prime \prime}(\phi_c),
\end{equation}
where $O_F$ is the fluctuation operator.
According to (\ref{E4}), one  obtains the supersymmetric form \cite{K87,R95}

\begin{equation}
\label{E14}
V^{\prime\prime}(\phi_c) = {U^\prime}^2 (\phi_c) +
U(\phi_c)U^{\prime \prime}(\phi_c),
\end{equation}
where the primes stand for a second derivative with respect to the
argument.

If the normal modes of (\ref{E13}) satisfy $\omega_n{^2} \geq 0,$  the
stability of the Schr\"odinger-like equation is ensured. Now, we are able
to implement a method that provides a new potential from  the potential
term that appears in the fluctuation operator.

Next, we consider the following generalized potential as corresponding to
the potential part of the fluctuation operator:

\begin{equation}
V^{\prime \prime}(\phi_c)=V(x;q) =
m^2(2-3q)sec^2h_q\left(\frac{m}{\sqrt 2}x \right),
\end{equation}
where $q>0$ and we are using the $q$-deformed hyperbolic functions which were
introduce by Arai \cite{arai}:


\begin{eqnarray}
\label{qHF} cosh_q(x)&&=\frac{e^x+qe^{-x}}{2}\nonumber\\
sinh_q(x)&&=\frac{e^x-qe^{-x}}{2}\nonumber\\
tanh_q(x)&&=\frac{sinh_q(x)}{cosh_q(x)}\nonumber\\
sech_q(x)&&=\frac{1}{cosh_q(x)}
\end{eqnarray}
where $x\epsilon${\bf R}. Thus

\begin{eqnarray}
\label{qHFd}
\frac{d}{dx}cosh_q(x)&&=sinh_q(x)\nonumber\\
\frac{d}{dx}sinh_q(x)&&=cosh_q(x)\nonumber\\ \frac{d}{dx}tanh_q(x)&&=
qsech_q^2(x)\nonumber\\
 \frac{d}{dx}sech_q(x)&&=
-tanh_q(x)sech_q(x)\nonumber\\
tanh_q^2(x)&&+qsech_q^2(x)=1.
\end{eqnarray}

The $q$-deformed potential term provides a fluctuation operator, so that their
eigenvalues satisfy the condition $\omega_n{^2}
\geq 0,$ and the ground state associated to the zero mode ($\omega_0^2=0$)
is given by

\begin{equation}
\label{GS} \eta^{(0)} (x;q) =Nsech^{2}_q\left(\frac{m}{\sqrt 2}x \right),
\end{equation}
where $N$ is the normalization constant. Thus, the stability of the
Schr\"odinger-like equation is ensured.

The potential model we are now going to study presents translational
invariance, then, the bosonic zero-mode eigenfunction of the stability
equation is related with the kink by

\begin{equation}
\label{MZ} \phi_q(x)= \int^{x}_{0}\eta^{(0)} (y;q)dy,
\end{equation}
so that, a priori, we may find the static classical configuration by a
first integration. Therefore, the potential model


\begin{equation}
\label{qp}
 V(\phi;q)=\frac 12\left(\frac{d}{dx}\phi(x;q)\right)^2
\end{equation}
yields a class of $q$-deformed scalar potentials, $V(\phi)=V(\phi;q),$
which have exact
solutions.

Expressing the position coordinate in terms of the kink, i.e.
$x=x(\phi_k),$ then, from (\ref{GS}) and (\ref{MZ}) we obtain the $q$-deformed
kink


\begin{equation}
\label{K}
\phi (x;q) = \frac{m}{\lambda q^2}tanh_q\left(\frac{m}{\sqrt 2}x
\right).
\end{equation}
The explicit form of the $q$-kink for few values of $q$ is
shown in Fig. 1.

From Eqs. (\ref{qp}) and (\ref{K}), we find a $q$-deformed
$\phi^4-$potential model with spontaneously
broken symmetry in scalar field theory, viz.,

\begin{equation}
\label{PD}
V(\phi;q) = \frac{\lambda^2}{4q^4}\left(q^2\phi^2 -
\frac{m^2}{\lambda^2}\right)^2.
\end{equation}
It represents a well-behaved potential energy. Note that the $q$-deformed
$\phi^4$ model has a discrete symmetry as $\phi\rightarrow-\phi$ but it is
spontaneously broken for the vacuum state by the existence of two
degenerate vacua:

\begin{equation}
\label{V} \phi_{1}= \frac{m}{q\lambda}, \qquad \phi_{2}= -\frac{m}{q\lambda}.
\end{equation}


The fact that the energy is finite is ensured because the kink
by the behavior of the approaches
one of the vacuum solutions at $\pm\infty.$ In the $q$-deformed
$\phi^4$ model there are four topological sectors, which are represented
by two
spaces $\Gamma_1$ and $\Gamma_2$ containing the $q$-deformed vacuum
solutions $\phi_1$
and $\phi_2$ and two spaces $\Gamma_3$ and $\Gamma_4$ containing the kink
and the anti-kink solutions.

The energy density of the $q$-kink is given by


\begin{equation}
\label{DE}
\epsilon(x)=\frac 12 \left[\left(\phi^{\prime}\right)^2+U^2\right]=
\frac{m^4}{2q^2\lambda^2}sech_q^4\left(\frac{m}{\sqrt 2}x \right).
\end{equation}


Therefore, the kink mass or the so-called classical mass of the
pseudoparticle is given by

\begin{equation}
\label{MK}
M_{cl} = \int_{-\infty}^{+\infty}\epsilon(x)dx
=\frac 23\left(\frac{m^3\sqrt{2}}{\lambda^2q^4}\right)
\end{equation}
which is dependent of $q.$ Note that when $q=1$
the undeformed case is re-obtained. In figure 2, we plot
the energy density given by
Eq.(\ref{DE}),
for few values of $q$.

The  conserved topological
current becomes:

\begin{equation}
\label{ct}
{\cal J}_{\mu}=\frac 12 \epsilon_{\mu\nu}\partial^{\nu}{\tilde\phi}_q, \quad
 {\tilde\phi}_q=\frac{m}{q\lambda}\phi_q, \quad
\partial^{\mu}{\cal J}_{\mu}=0,
\end{equation}
where $ \epsilon_{\mu\nu}$ is the antisymmetric tensor in two dimensions
$ \epsilon_{01}=- \epsilon_{10}=1$ and is zero for the case
with repeated index.
The kink number or conserved topological charge is given by

\begin{equation}
\label{carga}
Q=\int_{-\infty}^{+\infty}{\cal J}_{0}dx=\frac 12 [\lim_{x\rightarrow +\infty}
\tilde\phi_q(x)-
 \lim_{x\rightarrow -\infty}\tilde\phi_q(x)],
\end{equation}
which does not generate symmetries of the Lagrangian density and, therefore,
$Q$ is not a Noether charge. However, this charge is absolutely conserved,
$\frac{d}{dt}Q=0,$ so that the $q-$kink represents stable particle-like states.
Thus, the $q$-kink states can not decay by quantum tunneling into the
vacuum.

From the $q$-deformed potential, one then obtains the supersymmetric form

\begin{equation}
\label{FS}
V_-(x;q)
= W_q^2(x) + W^{\prime}_q,
\end{equation}
where the prime mean a first derivative with respect to the argument,
and $W_q(x)= -U^{\prime}_q(\phi_k)$ is the $q$-deformed superpotential
associated to the $q$-kink solution.
Thus, the bosonic and fermionic sector fluctuation operator
are respectively given by
\begin{eqnarray}
\label{E 19}
O_{F-} &&= -\frac{d^2}{dx^2}+W^2_q-W^{\prime}_q\nonumber\\
O_{F+} &&= -\frac{d^2}{dx^2}+W^2_q+W^{\prime}_q,
\end{eqnarray}
where $W_q(x) = -2mtanh_q\left(\frac{m}{\sqrt 2}x \right).$
These fluctuaction operators  are also called the supersymmetric partners,
which are isospectral up to the ground state.
The shape invariance condition of the pair of SUSY partner
potential will be investgated in a forthcoming paper.

\section{Conclusion}

In conclusion, we can say that, starting from a potential $V(x;q)$
in terms of the $q$-deformed
hyperbolic functions in the stability equation,
we construct the $q$-deformed topological kink associated
to the $q$-deformed $\phi^4$ potential model. We shown that the $q-$kink
mass  is dependent of $q.$
We stress that a very rich spectrum of the states
(the $q-$kink and the quantum excitations about them), which was totally
unexpected in this model has emerged because of the existence of soliton
solutions.
Finally, it is important to pointed out that one can extend our approach to 3+1 dimensions.
Indeed, the present work opens a
new route for future investigations on
domain walls \cite{vacha02} from $q-$deformation of potential model in terms
of coupled scalar fields. For instance, let us point out
that  our approach can be
applied from two \cite{Shif01b,RV96,guila02a} and three \cite{RPV95}
coupled scalar fields in higher dimensions, where in both cases the
$q-$soliton solutions only depend
on $z$ but not on $x$ and $y$ \cite{dlc02}.


\vskip 1.0 true cm

\noindent{{\large\bf Acknowledgments}}


RLR wishes to thanks the staff of CBPF and DCEN-CFP-UFPB for facilities.
This research was supported in part by CNPq (Brazilian Research Agency).
RLR is grateful to Prof. J. A. Helayel-Neto and
Prof. M. A.
Rego-Monteiro for fruitful discussions.
 The authors would also like to thank A. Arai and S.-C. Jia for the
kind interest in pointing out relevant
references on the subject of this paper.



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%%%%%%%%%% Figura 1

\unitlength=1cm
\begin{figure}[tbp]
\centering
\begin{picture}(10,1)
\epsfig{file=q-kink01.eps,width=10cm,height=08cm,angle=-90}
\end{picture}
\vspace{7.5cm}
\caption{The $q$-deformed kink profile,
with $q=0.8$(thickness=1), $q=1.0$(dotted curve), and $q=3.0$(thickness=3),
respectively, for $m=\lambda=\sqrt{2}.$
}
\end{figure}

%%%%%%%%%% Figura 2

\unitlength=1cm
\begin{figure}[tbp]
\centering
\begin{picture}(10,1)
\epsfig{file=qde101.eps,width=10cm,height=08cm,angle=-90}
\end{picture}
\vspace{7.5cm}
\caption{The energy density given by Eq.(\ref{DE}),
with $q= 0.8$(thickness=1), $q=1.0$(dotted curve), and $q=3.0$(thickness=3),
respectively, for $m=\lambda=\sqrt{2}.$}
\end{figure}


\end{document}

