\section{ Introduction}

A topological insulator is a material that behaves like an insulator in bulk, but on the surface electrons 
 can conduct electric current as in an ordinary conductor. On the surface of the insulator, the carriers are 
 Dirac fermions, but the conductivity depends on topological properties \cite{haldane}.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      
Notably, these systems show similar behavior to the FQHE but  an external magnetic field is  unnecessary. The similarity with the FQHE is that the carriers also live in a two-dimensional edge similar to the FQHE with anyonic behavior if the sample is three-dimensional and as chiral bosons if the insulator is two-dimensional \cite{wen} (see also \cite{moore,hansson}). 

From this point of view,  the analogy with the FQHE can be taken even further if we think of a topological insulator
 as a Hall hetero-junction where the magnetic field is not present, and the relevant question is, what is the analog of the 
 external magnetic field?

This problem was first studied by Kane and Mele \cite{kane1},  who assumed that in the absence of an external magnetic field, the spin-orbit term plays this role (see also \cite{fu,moore1,roy}).

In this paper, we would like to study the problem posed by Kane and Mele \cite{kane1} using supersymmetric quantum mechanics techniques  \cite{witten,review}. The advantages of solving this problem in this way are at least three: the first is that the spin-orbit coupling results from supersymmetrization and is not a non-relativistic correction. Secondly, using this procedure, we can  see how the $SU(2)$ connection emerges in the Hamiltonian, and third, the proposed  procedure
 might allow one to analyze other classes of models of topological insulators.

 In $d=3$,  there are at least two different ways to supersymmetrize a physical system;  the first one is based on Clifford's 
 algebra for fermionic fields \cite{witten},  and the other,  on a modification that induces a spin-orbit coupling term for the
  Hamiltonian \cite{jj} (see also \cite{gozzi} for the one-dimensional case).  This last method of  supersymmetrization is the 
  one we will use in this paper. 
 
 The paper is organized as follows; in section II, we  show the main ideas of this supersymmetry in  connection
 with the topological insulators.  In section III, we provide  two examples of how these ideas apply for the case of 
 a harmonic oscillator and the Aharonov-Bohm potentials; for the case of the oscillator in three dimensions, it turns
  out to be an extension of the system considered by Benevig and Zhang \cite{zhang}, while for the second we show how we can build a topological insulator using the Aharonov-Bohm effect. In the  last section contains the conclusions and  the discussion of the results.
 
 
\section{Topological insulators from supersymmetry}

Let us summarize the main ideas of the supersymmetrization procedure  discussed in \cite{jj}.  For any 
quantum mechanical  system with Hamiltonian \footnote{In what follows, Hamiltonians are multiplied by the 
mass $m$, but we preserve the standard notation $H$.}
\begin{equation}
H = -\frac{1}{2} \nabla^2 + u({\bf x}), 
\label{01}
\end{equation}
where $u({\bf x})$ is the potential, the ground state has no nodes and, therefore,  this particular state  can be written as 
\begin{equation} 
\psi_0=  e^{-V}, \label{f0}
\end{equation} 
where $V({\bf x})>0$, and  real.

Then, the Schr\"odinger equation for the ground state,  $H \psi_0 = E_0 \psi_0$,  reads 
\begin{equation} 
{\bar H} \psi_0 =0, \label{f1}
\end{equation} 
with ${\bar H} = H-E_0$.

Replacing (\ref{f0}) in (\ref{f1}), the Hamiltonian ${\bar H}$ can be written as follows 
\begin{equation}
{\bar H}= \frac12\,Q^\dagger_i Q_i,
\label{f2}
\end{equation}
with $\{i,j\}\in\{1,2,3\}$ and  
\begin{equation}
Q^\dagger_i = -\partial_i + \partial_i V, ~~~Q_i= \partial_i + \partial_i V,
\label{f3}
\end{equation}
and $ u=\frac12\left((\nabla V)^2 - \nabla^2 V\right)$.


Once $ {\bar H} $ is in form (\ref{f2}), it can be supersymmetrized following the 
prescription
\begin{eqnarray} 
Q_i &\to& S \,\,= Q_i \psi_i,  \nonumber
\\
Q^\dagger_i &\to& S^\dagger = Q^\dagger_i \psi^\dagger_i,   
\label{f4}
\end{eqnarray}
with $\psi_i$ and $\psi_j^\dag$, elements of a Clifford algebra. While for the one dimensional 
case these elements  satisfy  $\{\psi,\psi\}=0=\{\psi^\dagger, \psi^\dagger\}$, $\{\psi,\psi^\dagger\}=1$,  
in higher dimensions  this  structure may be  more complicated. Indeed, in  three dimensions
\begin{equation}
\{\psi_i,\psi_j\} =0=\{\psi^\dagger_i, \psi^\dagger_j\},\quad
\{\psi_i, \psi^\dagger_j\}  =  \delta_{ij} +  i \kappa_{ij}
\label{f5}
\end{equation}
where $\kappa_{ij}=-\kappa_{ji}$ and $\kappa^\dagger =\kappa$.  In the present case, 
the $\{\psi_i\}_{i=1,2,3}$ defined as
\begin{equation}
\psi_i= \sigma_i \otimes \sigma_+,
\quad
\psi^\dagger_i =\sigma_i \otimes \sigma_-, \label{h1}
\end{equation}
where ${\sigma_i}$ are the Pauli matrices and $\sigma_{\pm} =\frac12(\sigma_1\pm\imath\sigma_2)$,
furnish a realization of the previous algebra with 
\begin{equation}
\kappa_{ij} = \epsilon_{ijk}\sigma_k \otimes \sigma_3. \label{ff5}
\end{equation}

For $\kappa_{ij}= 0$, the spinors $\psi_i, \psi_j^\dagger$ obey a standard Clifford algebra, but if $\kappa_{ij} \neq 0$, the grading of the algebra is  more involved  \cite{balantekin,kost}. For harmonic-like potentials and their variations with centrifugal potentials this is a well-established fact. The problem is more intricate for general central potentials, as was discussed in   \cite{ritte}.

The prescription in   (\ref{f4}), with $\kappa_{ij}$  given in (\ref{ff5}), generates the superalgebra
\begin{eqnarray}
\{S,S^\dagger\} &=& 2 {\bar H}_s, \label{g1}
\\
\left[ S, {\bar H}_s\right] &=& 0=\left[S^\dagger, {\bar H}_s\right], \label{g22}
\\
\{S^\dagger, S^\dagger\} &=&0=\{S,S\}. \label{g3}
\end{eqnarray}
where \footnote{We will suppress the symbol $\,\bar{ }\, $ in the rest of the paper and it is understood that the 
Hamiltonians under discussion are shifted so that the energy of the ground state is zero.}
\begin{equation}
{\bar H}_s =
H_{-}\otimes \sigma_- + H_{+} \otimes \sigma_+, \label{g4}
\end{equation}
with
\begin{eqnarray}
H_{\pm} &=& -\frac{1}{2}\nabla^2 + u_\pm(\textbf{x}) \pm 2(\nabla V \times {\bf p}) \cdot\,\textbf{s}, 
\label{g5}
\\
u_{\pm}(\textbf{x}) &=& \frac{1}{2}[(\nabla V)^2 {\pm} \nabla^2 V]. \label{p1}
\end{eqnarray}
where  ${\bf s} =\frac12{\boldsymbol \sigma}$.
 The appearance of the spin-orbit coupling term ensures  the invariance under $T$ and introduces a magnetic field-like 
 term. To see this,  we will proceed as follows; let us  take,  for example, the component $H_{-}$ in (\ref{g5})  and 
 write it in the following suggestive form
 \begin{eqnarray}
 H_{-} 

 

 &=& \frac{1}{2}\left[{\bf p} -2\,{\bf s}\times \nabla V \right]^2 -2\left( {\bf s}\times \nabla V\right)^2 + u_-(\textbf{x}), \label{p111}
 \end{eqnarray} 
 where we have used $(\nabla V\times {\bf p})\cdot{\bf s} =({\bf s}\times\nabla V)\cdot{\bf p}$, and $
 [{\bf s}\times\nabla V,{\bf p}] =0$.  Since  $\left( {\bf s}\times \nabla V\right)^2=\frac{1}{2}(\nabla V)^2$, 
 we rewrite (\ref{p111}) as follows
\begin{equation}
H_{-} = \frac{1}{2}\left[{\bf p} -2\,{\bf s}\times \nabla V \right]^2 + \bar{u}_-({\bf x}),
\end{equation}
with
\begin{equation}
 {\bar u}_{-}({\bf x})=  -\frac{1}{2} \nabla^2 V.
 \label{pp2}
 \end{equation} 
By  changing $V \to -V$ (and then $\bar{u}_-\to \bar{u}_+$), we obtain $H_{+}$.

To summarize,  it is possible to write $H_{\pm }$ defined in (\ref{g5})  as 
\begin{equation}
\label{hmag}
H_{\pm } = \frac{1}{2} \left[ {\bf p} \pm {\bf {\cal A}}\right]^2 +\bar{u}_{\pm}({\bf x}),
\end{equation}
with ${\bf {\cal A}} = 2{\bf s}\times \nabla V$.


 
 We emphasize that the term ${\bf {\cal A}}= 2{\bf s}\times \nabla V$  turns out to be a $SU(2)$ connection. Indeed,
 let us define ${\cal A}^a_i={\epsilon_i}^{a k}\partial_kV$, and therefore
 \footnote{For this part of the discussion we have used a modified notation where the group index is denoted by latin
index ${a,b,\dots}$.} ${\bf {\cal A}}_i ={{\cal A}^a}_i\,\sigma^a$ is an element of the $SU(2)$ algebra.  In order ${\bf {\cal A}}$ to be a connection, it must change under the $SU(2)$ local transformation  $U=e^{\imath \theta^a\sigma^a}$ as
\begin{equation}
{\bf {\cal A}}_i\to{\bf {\cal A}'}_i=U\,{\bf{\cal A}}_i\,U^{\dag}+\imath\,\partial_iU\,U^{\dag}.
\end{equation}
with $U^{\dag} =U^{-1}$.

Global gauge transformations are particularly interesting for our model. In fact, for $\theta^a$ constant, the  gauge transformation reads
$$
{\bf {\cal A}}\to{\bf {\cal A}'}=U\,{\bf{\cal A}} \,U^{\dag},
$$
and then, we perform the unitary transformation ${\bm {U}} =U \otimes \openone$ in $\bar{H}_s$ defined in  (\ref{g4}) as 
follows
\begin{eqnarray}
\bar{H}'_s &=&{\bm U}\,\bar{H}_s\,{\bm U}^\dag,
\nonumber
\\
&=&U\,H_{+}\,U^{\dag}\otimes\sigma_+\,+\,U\,H_{-}U^\dag\otimes\sigma_-,
\nonumber
\\
&\equiv& H'_{+}\otimes \sigma_+  \,+\,  H'_{-}\otimes\sigma_-.
\end{eqnarray}
 
Since $U({\bf p} -{\bf {\cal A}})^2\,U^\dag = ({\bf p} - U{\bf {\cal A}}U^\dag)^2$, the unitary transformed Hamiltonians
are
\begin{equation}
H'_{\pm } =  \frac{1}{2} \left[ {\bf p} \pm {\bf {\cal A}}'\right]^2 +\bar{u}_{\pm}({\bf x}).
\end{equation}
 
 
For example, consider de gauge field ${ {\cal A}}_i=A_i\, n^a \sigma^a =A_i\,{\bf n}\cdot{\boldsymbol \sigma}$, with 
${\bf n}$ a fixed direction in the $SU(2)$ algebra. The transformation

 



\begin{align}
U &= U_3[\varphi/2]\,U_2[\theta/2]U_3[-\varphi/2],
\nonumber
\\
&=e^{\imath \frac{\varphi}{2}\sigma_3}\,e^{-\imath \frac{\theta}{2}\sigma_2}\,e^{-\imath \frac{\varphi}{2}\sigma_3}
\nonumber
\\
&=\left( \begin{array}{lr} 
\cos \frac{\theta}{2} & -e^{i\varphi} \,\sin \frac{\theta}{2} 
\\
e^{-i\varphi}\,\sin \frac{\theta}{2}  & \cos \frac{\theta}{2}
\end{array} \right),
\nonumber
\end{align}
and the identity (Hopf map) \cite{aitchi}
\[
U \,{\hat {\bf n}}\cdot\mbox{\boldmath$\sigma$}\,U^\dagger = \sigma_3, 
\]
allow to transform equation (\ref{g4}) to
\begin{equation}
{\bar H}_s'= {\bar H}'_{-}\otimes \sigma_- + {\bar H}'_{+} \otimes \sigma_+, \label{g41}
\end{equation} 
where 
\begin{equation}
{\bar H}'_{\pm } =[ -\frac{1}{2}\nabla^2 + {u}_\pm(\textbf{x})\pm (\nabla V\times {\bf p})_3\,\sigma_3]. 
\label{g511}
\end{equation}

For the case of a central potential $V({\bf x})=V(r)$, one has 
$$ \nabla V\times{\bf p} = \frac{dV}{dr}\hat{r}\times{\bf p}
=\frac{1}{r}\frac{dV}{dr}(r\hat{r}\times{\bf p})=\frac{1}{r}\frac{dV}{dr}{\bf L},
$$
with ${\bf L}$, the angular momentum.

Therefore, in the present case, after the gauge transformations, we get for the central potential
\begin{equation}
{\bar H}'_{\pm} =[ -\frac{1}{2}\nabla^2 + {u}_\pm(\textbf{x}) \pm \frac{L_3}{r} \frac{dV}{dr}  \sigma_3], 
\label{g5111}
\end{equation}
and since $[L_3,H'_{\pm}]=0$,  operator  $L_3$  can be replaced by its eigenvalues.

On the other hand, in this approach, we can define the covariant derivative and the curvature in the standard way, and 
since ${\cal A}$ is a non-abelian connection, the curvature ${\cal F}_{\mu \nu}$ is not gauge invariant. However, for a topological insulator, the invariance under $T$ allows to invert the momentum and the spin simultaneously and, therefore, if we choose the spin on the axis-$3$, the two components of the constant chromomagnetic field \cite{weis} can be interpreted as a real magnetic field in the up and down directions.


The curvature can be calculated from the commutator $\left[D_i({\cal A}),D_j({\cal A})\right]$, 
\begin{equation}
\left[D_i({\cal A}),D_j({\cal A})\right] = \imath {\cal F}_{ij}, 
\end{equation}
with $ D_i({\cal A}) = \partial_i + 2i ({\bf s}\times \nabla V)_i$ (and a similar definition for $H'_{+}$, by changing $V\to -V$)



 \[
 {\cal F}_{ij} = 2\imath\left(\epsilon_{jkl} \partial_i \partial_l V -\epsilon_{ikl} \partial_j \partial_l  V \right) s_k.
 \]

Therefore, using the arguments explained above, for the connection along $\sigma_3$, one finds
\begin{equation}
{\cal F}_{12}=\mathbb{B}= \nabla^2 V\, \sigma_3, \label{spin}
\end{equation}
where $\nabla^2$ is the two-dimensional Laplacian and the eigenvalues of ${\cal F}_{12}$ contains the up and down components for the \lq \lq magnetic field\rq \rq \,stated above.

From these results we see that  Hamiltonian 
\begin{equation}
{\bar H}_s=\left[\frac{1}{2} ({\bf p} -{\cal A})^2 + {\bar u}_-({\bf x}) \right] \otimes \sigma_- + \left[\frac{1}{2} ({\bf p} +{\cal A})^2 + {\bar u}_+({\bf x}) \right] \otimes \sigma_+, \label{hh2}
\end{equation} 
describes a supersymmetric system coupled to a $SU (2)$ gauge field \cite{wong,bal} and the role played by this field is to implement the topological magnetic field. Note that in three dimensions --as in anyons where one adds a Chern-Simons term in $2+1$ dimensions \cite{rich}-- we should add a Pontryagin term, but this is a total derivative in the Lagrangian and does not contribute to the Hamiltonian. However, the total derivative leaves a non-trivial topological effect;  it is responsible for the magnetoelectric effect.

Note also that,    if   $\nabla V \times {\bf p}$ varies adiabatically, then we can approximate 
\begin{equation}
{\bar H}_{\pm } = \frac{1}{2}{\bf p}^2 + {u}_{\pm}({\bf x}) \pm 2 (\nabla V \times {\bf p})\cdot{\bf s}, \label{gp1}
\end{equation}
to
\begin{eqnarray}
{H}_{\pm } &\simeq &- 2(\nabla V \times {\bf p})\cdot{\bf s} \nonumber
\\
&\simeq& - {\bf B}\cdot{\bf s}, \label{gp2}
\end{eqnarray}
where ${\bf B}=2\nabla V \times {\bf p}$ is interpreted as a  fictitious magnetic field. What does this approximate physically mean?

Let us assume that over  some typical length scale ${\it L}$ the following relation holds $ |\partial_i V|\gg L\,\partial_i\partial_j V$, (in one dimension this means $V'\gg LV''$)
so that the potential is of the order $V\approx V_0 +L|\nabla V|$, with $V_0$ the mean value of the potential inside the region of size $L$. In this case, the potential $u_\pm\sim (\nabla V)^2$.  For heavy modes, that is $p^2\ll m$, the condition 
$|\nabla V|\ll |{\bf p}|$, assures that the term $|\nabla V\times {\bf p}|$ dominates the complete Hamiltonian, giving rise
to the adiabatic limit previously discussed.  


Using the de Broglie relation we can write an energy scale $E_{s}\sim \frac{1}{\lambda}$, characterizing the slow modes 
(highly massive modes) and also the energy scale $E_{f}\sim \frac{1}{L}$, characterizing the energy necessary to explore 
the region of size $L$. The condition $|\nabla V|\ll |{\bf p}|$, reads now 
\begin{equation}
\frac{E_f}{E_s} \ll1.
\end{equation}

The adiabatic approximation and its topological implications for this kind of problems
was studied by Berry \cite{berry} who discovered that in the parameter 
space ${\bf B }$ the gauge potential of this system is a Dirac monopole described by 
\begin{equation}
{\cal A}_i = \frac{1}{2}\frac{B_i}{|{\bf B}|^3}. \label{gp3}
\end{equation}
where the Berry phase $\gamma_{\pm}(C)$ satisfies
\begin{equation} 
e^{i \gamma_{\pm} (C)}= e^{\oint_C dB_i {\cal A}_i} = e^{\mp i {\frac{n}{2} \Omega (C)}},
\end{equation}
where $\Omega$ is the solid angle subtended in the parameters space.
\section{Examples Topological insulators  Quantum mechanics }

In order to illustrate the ideas set out above, we will discuss two examples of topological insulators that can be studied directly using our arguments.
\subsection{Harmonic oscillator in three dimensions} 

This example is a  variation of the problem studied by Benevig and Zhang \cite{zhang}. Consider
 the three-dimensional harmonic oscillator with frequency $\omega$ and mass $m$. The ground state (up to 
 a normalization constant)  is
\begin{equation}
\psi_0 =  e^{-\frac{m \omega}{2} r^2},
\end{equation} 
with $r$ the radial coordinate. The function $V$ in  (\ref{f0}) reads  $V= \frac{m \omega}{2} r^2$, and 
the potential in  (\ref{p1}) is 
\begin{equation}
u_\pm = \frac{1}{2}\left( m^2 \omega^2 r^2\mp 3m \omega \right). 
\label{pot1}
\end{equation}
For the supersymmetric  Hamiltonian ${\bar H}'_s$ in (\ref{g41}) (restoring the mass $m$), the ${\bar H}'_\pm$ components
(see  (\ref{g5111})) read
\begin{equation}
{\bar H}'_\pm =-\frac1{2m}\nabla^2+\frac{1}{2}\left( m \omega^2 r^2\mp 3 \omega \right)\pm\,\omega\,L_3\sigma_3,
\end{equation}
and  the Schr\"odinger-Pauli  equation  turn out to be
 \begin{equation}
 {\bar H}'_\pm\psi_\pm=E\,\psi_\pm.
 \end{equation}
Let us consider the diagonalization of ${\bar H}'_-$, that is
$$
 {\bar H}'_-\psi_-=E\,\psi_-.
$$

 We look for solutions for the spinor $\psi_-$ with the form
 \begin{equation}
\label{soloscp}
\psi_-(r,\theta,\varphi) =
\left(
\begin{array}{c}
R_-^{(+)}(r) Y_\ell^{n^{(+)}}(\theta,\varphi)
\\
R_-^{(-)}(r) Y_\ell^{n^{(-)}} (\theta,\varphi)
\end{array}
\right),
\end{equation} 
 where $Y_\ell^n(\theta,\varphi)$ 
  are the spherical harmonics. 
  
  The equation for $R_{-}^{(\pm)}(r)$ turn out to be
  \begin{equation}
  \label{oscp}
  {R_{-}^{(\pm)}}''+\frac{2}{r}{R_-^{(\pm)}}' 
  -\left[\frac{\ell(\ell+1)}{r^2} + m\omega\left(m\omega\,r^2+3\right) -2mE\mp2m\,n^{(\pm)}\omega\right]{R_-^{(\pm)}}=0.
  \end{equation}
In terms of a dimensionless variable $x=\sqrt{m\omega}\,r$, and defining $\epsilon^2 =\frac{2E}{\omega}$, 
previous equation reads
\begin{equation}
\label{osc2}
x^2\,{R_-^{(\pm)}}''+2x\,{R_-^{(\pm)}}'-\left[x^4+(3-\epsilon^2\mp2n^{(\pm)})x^2+\ell(\ell+1)\right]{R_-^{(\pm)}}=0,
\end{equation}
whose normalizable solutions  are
\begin{equation}
{R_-^{(\pm)}}(x) = x^\ell\,{e^{-\frac{x^2}{2}}}\,L^{\ell+1}_{\kappa^\pm}(x),
\end{equation}
with $2\kappa^{(\pm)} = 3+\frac{E}{\omega}-\ell\pm n^{(\pm)}$.  The solution is square integrable for $\kappa^{(\pm)}\in \mathbb{Z}^+$. Finally, $\psi_-$ then reads (up to a normalization constant)
\begin{equation}
\label{soloscp}
\psi_-(r,\theta,\varphi) = r^\ell\,e^{-\frac{m\omega r^2}{2}}
\left(
\begin{array}{c}
L^{\ell+1}_{\kappa^{(+)}}(x)\,Y_\ell^{n^{(+)}}
\\
L^{\ell+1}_{\kappa^{(-)}}(x)\,Y_\ell^{n^{(-)}}
\end{array}
\right).
\end{equation}

Note that the energy levels  $E=\omega(\ell-3+2\kappa^{(\pm)}\mp n^{(\pm)})$ are well defined for $2\kappa^{(+)}
-n^{(+)}=2\kappa^{(-)}+n^{(-)}$ only. Condition $E>0$ imposes also $2\kappa^{(\pm)}>3-\ell\pm n^{(\pm)}$.

Consider the case for which $\kappa^{(+)}>\kappa^{(-)}$. Then, the previous restrictions imply $n^{(+)}+n^{(-)}>0$.
For the case $\kappa^{(+)}<\kappa^{(-)}$, restriction reads $n^{(+)}+n^{(-)}<0$. That is, the $L_3$ projections of
the spinor's components are not independent.

The situation for  
$\kappa^{(+)} =\kappa^{(-)}$, is similar and projections must satisfy $n^{(-)} = - n^{(+)}$. We can write the solution 
for this case as follows
\begin{equation}
\label{soloscpequ}
\psi_-(r,\theta,\varphi) = r^\ell\,{e^{-\frac{m\omega r^2}{2}}}
\left(
\begin{array}{c}
L^{\ell+1}_{\kappa}(x)\,Y_\ell^{n}
\\
L^{\ell+1}_{\kappa}(x)\,Y_\ell^{-n}
\end{array}
\right).
\end{equation}

Clearly, in all three cases, the energy can be written as $E_{\kappa^{(+)},\ell,n^{(+)}}=\omega(2\kappa^{(+)}+\ell-3-n^{(+)})=\omega(2\kappa^{(-)}+\ell-3-n^{(-)})$, and the system is degenerated. It is worthwhile to notice that in the  
absence of the term proportional to $L_3$, the solution is $n^{(\pm)}$-independent and therefore $\kappa^{(+)}=
\kappa^{(-)}$ so that the solution depends only on integers $\kappa$ and $\ell$.

The diagonalization of ${\bar H}'_+$ proceeds in a similar way. We look for a solution for the  spinor $\psi_+$ 
with the form 
\begin{equation}
\label{soloscm}
\psi_+(r,\theta,\varphi) =
\left(
\begin{array}{c}
R_+^{(+)}(r) Y_\ell^{n^{(+)}}(\theta,\varphi)
\\
R_+^{(-)}(r) Y_\ell^{n^{(-)}} (\theta,\varphi)
\end{array}
\right).
\end{equation} 
Radial components are solutions of 
  \begin{equation}
  \label{oscm}
  {R_+^{(\pm)}}''+\frac{2}{r}{R_+^{(\pm)}}' 
  -\left[\frac{\ell(\ell+1)}{r^2} + m\omega\left(m\omega\,r^2-3\right) -2mE \pm 2m\,n^{(\pm)}\omega\right]{R_+^{(\pm)}}=0,
  \end{equation}
or, in terms of dimensionless variables
\begin{equation}
\label{osc2}
x^2\,{R_+^{(\pm)}}''+2x\,{R_+^{(\pm)}}'-\left[x^4+(-3-\epsilon^2 \pm 2 n^{(\pm)})x^2+\ell(\ell+1)\right]{R_+^{(\pm)}}=0.
\end{equation}
Solutions of this equations are those in (\ref{soloscp}) while the energy turn out to be $E=\omega(2\kappa^{(\pm)}\pm n^{(\pm)} +\ell)$, from which we obtain the condition  $2\kappa^{(+)}
+ n^{(+)}=2\kappa^{(-)}-n^{(-)}$, while  $E>0$ imposes also $2\kappa^{(\pm)}+\ell>\mp n^{(\pm)}$.


 \subsection{ The Aharonov-Bohm Topological Insulator}
  
 A second example is the case of the Aharonov-Bohm effect, which we can study directly through the identification of 
 $V$ according to 
 \begin{equation}
 \nabla^2 V = \alpha\, \delta ({\bf x}_\perp), 
  \end{equation}
 where $\alpha$ is a dimensionless coefficient defined in terms of the magnetic flux as  $\Phi_B=2\pi\alpha$ and 
 ${\bf x}_\perp$ are coordinates in two dimensions. 



In order to calculate $u_\pm$ we need to solve
\begin{equation} 
\nabla^2 V=  \frac{\Phi_B}{2\pi} \delta ({\bf x}_\perp),
\label{mono1}
\end{equation}
what can be achieved, for example, by using the Green function $G(|{\bf x}-{\bf x}'|)$. Indeed,
\begin{eqnarray}
V({\bf x}_\perp) &=& \frac{\Phi_B}{2\pi} \int  \, G(|{\bf x}_\perp-{\bf x}'_\perp|)\delta^{(2)} ({\bf x}')\,d^2x'
 \nonumber
\\
&=&\frac{\Phi_B}{2\pi} \,G(|{\bf x}_\perp|), 
\label{mono2}
\end{eqnarray}
with 
\begin{equation}
G(|{\bf x}_\perp|) = \int \frac{d^2 p}{(2\pi)^2} \frac{e^{i {\bf p}\cdot{\bf x}_\perp}}{{\bf p}^2} = \ln |\mu  {\bf x}_\perp|, \label{mono3}
\end{equation}
with $\mu$ an energy scale.

Using (\ref{p1}) and (\ref{mono2}) the potential  is
\begin{equation}
u_\pm ({\bf x}_\perp) = \frac{1}{2} \left[ \frac{\Phi_B^2}{ 4 \pi^2|{\bf x}_\perp|^2} \pm  \frac{\Phi_B}{2\pi} \delta ({\bf x}_\perp)\right], \label{mono4}
\end{equation}
and the Hamiltonian ${\bar H}'_\pm$ in  (\ref{g5111}) (or ( \ref{g5}))  becomes 
\begin{equation}
{\bar H}'_\pm = \frac{1}{2} \left[ -\nabla^2 + \frac{\Phi_B^2}{4 \pi^2|{\bf x}_\perp|^2} \pm \frac{\Phi_B}{2\pi}~\delta ({\bf x}_\perp) \right] \pm \frac{\Phi_B}{2 \pi |{\bf x}_\perp|^2} \sigma_3 L_3. \label{last}
\end{equation}

The diagonalization of the supersymmetric Hamiltonian implies, as before, the following equations
\begin{equation}
{\bar H}'_\pm\,\psi_\pm=E\psi_\pm.
\end{equation}
Explicitly, in polar coordinates, this equation reads
\begin{equation}
{\bar H}'_\pm =\frac{1}{2} \left[-\nabla_\perp^2 - \partial_z^2 
+ \frac{\Phi_B^2}{4 \pi^2{\rho^2}}
 \pm 
 \frac{\Phi_B}{2\pi}\frac{\delta (\rho)}{\rho}\right]   \pm \frac{\Phi_B}{2 \pi \rho^2} \sigma_3 L_3.
\label{last1}
\end{equation}
with $\nabla_\perp^2=\rho^{-1}\partial_\rho[\rho\partial_\rho\,]+\rho^{-2}\partial_\theta^2$.

Let us consider first the ${\bar H}'_-$. For such a case we look for solutions with the form
\begin{equation}
\psi_-(\rho,\theta,z) =e^{\imath k z} \sum_{n=-\infty}^{n=\infty}\left(
\begin{array}{c}
R_-^{(+)}\,e^{\imath \theta n^{(+)}}
\\
R_-^{(-)}\,e^{\imath \theta n^{(-)} }
\end{array}
\right),
\end{equation}
where the  integers $n^{(\pm)}$ are the winding numbers.   

The radial part,  satisfies
\begin{equation}
{R_-^{(\pm)}}''+\frac1\rho{{R_-^{(\pm)}}'}-(k^2-\epsilon^2){R_-^{(\pm)}}-
\frac1{\rho^2}\left(n^{(\pm)} \mp \frac{\Phi_B	}{2\pi}\right)^2{R_-^{(\pm)}}=
 \frac{\Phi_B	}{2\pi}\frac{\delta(\rho)}{\rho}{R_-^{(\pm)}},
 \label{last1}
\end{equation}
with $\epsilon^2=2E$. For the Hamiltonian ${\bar H}'_+$, the radial part equation can be obtained from this one
by changing $\Phi_B\to-\Phi_B$.



The factor $\frac{\Phi_B}{2\pi}$ in (\ref{last1})   is the magnetic field inside the solenoid, while  the change of  signs 
in front the $\delta$-function when passing from ${\bar H}'_-$ to  ${\bar H}'_+$, shows that the system is a topological 
invariant (for a study from a different point of view see for example \cite{moo1,ab2}). The complete solution of the 
problem is the superposition of different $n^{(\pm)}$, that is, different  winding numbers for spin components up and down.


\section{ Final Comments}


The analysis of topological insulators in the context of quantum mechanics (and eventually quantum field theory) is an interesting possibility that can contribute new ideas in physics beyond the condensed matter physics. In this paper we have proposed a route on how to proceed.

The first conclusion that follows from our approach is that the need to include a spin-orbit term does not depend on the energy scale as can be seen using supersymmetry {\it a l\'a} spin-orbit,  and therefore,  a connection between supersymmetry and topology is a  consequence that emerges and will  require a more rigorous analysis in other applications. The second conclusion is that these results open the possibility that  other quantum mechanical systems could be analyzed from this perspective and their implications in noncommutative quantum mechanics and integrable quantum systems seem to be possible . 

Finally, the problem of extending the current approach to relativistic systems and
 variations around the Dirac like oscillator \cite{moshinsky} is a matter of current research.

 
\noindent 

\section*{Acknowledgements}

  One of us (J.G.) thanks Prof.  Andreas Ringwald for the discussions and hospitality at DESY (Hamburg) and the Alexander von Humboldt Foundation for the financial support. We would like also to thank H. Falomir and J. Zanelli for helpful discussions. This research was supported by DICYT 042131GR (J.G.) and 041931MF (F.M.). 
 
 
 
