#  A-geometrical approach to Topological Insulators with defects

D. Schmeltzer  Physics Department, City College of the City University of New
York
New York, New York 10031

###### Abstract

The study of the propagation of electrons with a varying spinor orientability
is performed using the coordinate transformation method.

Topological Insulators are characterized by an odd number of changes of the
orientability in the Brillouin zone. For defects the change in orientability
takes place for closed orbits in real space. Both cases are characterized by
nontrivial spin connections.

Using this method , we derive the form of the spin connections for topological
defects in three dimensional Topological Insulators.

On the surface of a Topological Insulator, the presence an edge dislocation
gives rise to a spin connection controlled by torsion. We find that electrons
propagate along two dimensional regions and confined circular contours. We
compute for the edge dislocations the tunneling density of states. The edge
dislocations violates parity symmetry resulting in a current measured by the
in-plane component of the spin on the surface.

I Introduction

The propagation of electrons in solids is characterized by the topological
properties of the the electronic band spinors. Topological Insulators Konig ;
Volkov ; Gotelman ; Kreutz ; Mele ; Kane ; More ; Essin ; BernewigZhang ;
Ludwig ; davidtop ; ZhangField ; Zhangnew  can be identified by an odd number
of changes of the $orientability$ davidtop  of the spinors in the Brillouin
zone. As a results non trivial spin connections with a non- zero curvature
characterized by the Chern numbers can be identified. In time reversal
invariant systems one finds that for Kramer’s states the time reversal
operator $T$ obeys $T^{2}=-1$ and one thus the second Chern number for four
dimensional space is given by $(-1)^{\nu}=-1$, where $\nu$ is an odd number of
orientability changes Nakahara  .

Real materials are imperfect and contain topological defects such as
dislocations Ran ; Sinova ,disclinations alberto ; Vozmediano  and gauge
fields induced by strain in graphene Baruch ; Guinea  ;therefore, a natural
question is to formulate the physics of Topological Insulators in the presence
of such defects davidtop . These topological defects can be analyzed using the
coordinate transformation method given in ref.kleinert  which modifies the
Hamiltonian for a Topological Insulator with a defect by the metric tensor and
the spin connection Pnueli ; Green ; Birrell ; Randono ; Ryu .

The effect of strain fields dislocations and disclinations plays an important
role in material science and can be study using Scanning Tunneling Microscopy
($STM$) and Transmission Electron Spectroscopy ($TEM$ ). Therefore we expect
that the chiral metallic boundary Wu  will be sensitive to such defects.

In this paper we will introduce the tangent space approach used in
differential geometry Nakahara ; Randono ; Ryu  to study propagation of
electrons for a space dependent coordinate kleinert . We find that the
continuum representation of the edge dislocation kleinert  generates a spin
connection Pnueli ; Green ; Birrell  which is controlled by the $Burger$
vector.

Using this formulation we obtain the form of the topological insulator in
three dimensions which simplifies for the surface Hamiltonian (on the
boundary). For the surface Hamiltonian we find that the electronic excitations
are confined to a two-dimensional region and to a set of circular contours of
radius $R_{g}(n)$.

The contents of this paper is as follows: In chapter $II$ we introduce the
gemetrical method. In section $IIA$ we present the geometrical method for the
edge dislocations and strain fields. In section $IIB$ we consider the effects
of the strain fields on the three- dimensional Topological Insulator ($TI$).
The Chiral model for the boundary surface is presented in section $IIIA$.
Section $IIIB$ is devoted to the derivation of the metric tensor and spin
connection for an edge dislocation kleinert . In section $IIIC$ we identify
the stable solutions. Section $IIID$ is devoted to the stable two dimensional
solutions $n=0$ and section $IIIE$ is devoted to the stable solution for
circular contours $n=\pm 1$. Chapter $IVA$ is devoted to the computation of
the tunneling density of states. In section $IVB$ we present results for the
two dimensional region $n=0$. Section $IVC$ is devoted to a large number of
dislocations. In section $IVD$ we compute the tunneling density of states for
the circular contours $n=\pm 1$. In chapter $V$ we consider the current which
is given by the in-plane spin component. In section $VA$ we show that this
current is zero for a $TI$. In section $VB$ we show that in the presence of an
edge dislocation the parity symmetry is violated, and current, representing
the in-plane spin component, is generated. Chapter $VI$ is devoted to
conclusions.

II-The Geometrical method for dislocations and strain fields

A-General Considerations

A perfect crystal is described by the lattice coordinates $\vec{r}=[x,y,z]$.
For a crystal with a deformation , the coordinates $\vec{r}$ are replaced by
$\vec{r}\rightarrow\vec{R}=\vec{r}+\vec{u}\equiv[X^{1}(\vec{r}),X^{2}(\vec{r}),X^{3}(\vec{r})]$
where $\vec{u}(\vec{r})$ is the local lattice deformation and $X^{a}$,
$a=1,2,3$ is the local coordinate which changes when we move from one point to
another.

In a deformed crystal we introduced a set of local vectors $e_{a}$ which are
orthogonal to each other $(e_{b},e_{a})\equiv<e^{b}|e_{a}>=\delta^{b}_{a}$ and
local coordinates $X^{a}$, $a=1,2,3$. The unit vector $e_{a}$ can be
represented in terms of a Cartesian fixed frame space with the  coordinate
basis $\partial_{\mu}$ ,$\mu=x,y,z$. In the fixed Cartesian frame the
coordinates are given by $x^{\mu}$. Using the Cartesian basis $\partial_{\mu}$
we expand the deformed medium in terms of the local tangent vector $e_{a}$ :
$e_{a}=e^{\mu}_{a}\partial_{\mu}$ (for the particular case where vectors
$e_{a}$ are given by $e_{a}=\partial_{a}$, the transformation between the two
basis is $e^{\mu}_{a}=\delta^{\mu}_{a}$). Any vector $\vec{X}$ (in the
deformed space) can be represented in terms of the unit vectors $e_{a}$ or the
$\partial_{\mu}$ (the tangent vectors in the Cartesian fixed coordinates
space). The vector $\vec{X}$ can be represented in two different ways,
$\vec{X}=X^{a}e_{a}=X^{\mu}\partial_{\mu}$ (when an index appear twice is
understood as a summation, $X^{a}e_{a}\equiv\sum_{a=1,2,3}X^{a}e_{a}$). The
dual vector $e^{a}$ is a $one$ $form$ and can be expanded in terms of the one
forms $dx^{\mu}$. We have: $e^{a}=e^{a}_{\mu}dx$, where $e^{a}_{\mu}$
represents the matrix transformation
$e^{a}\equiv(\partial_{\mu}X^{a})dx^{\mu}$. The scalar product of the
components $e^{a}_{\mu}e^{a}_{\nu}=g_{\mu,\nu}$,
$e^{\nu}_{a}e^{\nu}_{b}=\delta_{a,b}$ defines the metric tensors,
$g_{\mu,\nu}$ (in the Cartesian frame ) and $\delta_{a,b}$ in the local medium
frame.

B-Application to the Topological insulators in three dimensions

The three dimensional electronic $TI$ bands for $Bi_{2}Se_{3}$ and
$Bi_{2}Te_{3}$ can be represented using four projected states Chao ,
$|orbital=1,2>\otimes|spin=\uparrow,\downarrow>$ (the Pauli matrix $\tau$
describes the orbital states and the Pauli matrix $\sigma$ describes the
spin). The effective $h^{3D}$ Hamiltonian in the first quantized form is given
by:

$h^{3D}=\hbar
v_{0}[k_{y}(\sigma_{1}\otimes\tau_{1})-k_{x}(\sigma_{2}\otimes\tau_{1})+\epsilon
k_{z}(\sigma_{3}\otimes\tau_{1})+M(\vec{k})(I\otimes\tau_{3})]$ (1)

The parameter $M(\vec{k})$ determines if the insulator is trivial or
topological. For $Bi_{2}Se_{3}$ and $Bi_{2}Te_{3}$ the gap is inverted, namely
$M(\vec{k})=-M_{0}+B_{1}k_{z}^{2}+B_{2}k_{\bot}^{2}$ with
$M_{0}>0,B_{1}>0,B_{2}>0$ and therefore topological ZhangField ; Chao ;
Zhangnew .

Using the metric tensor $g_{\mu,\nu}$ given by the coordinate transformation (
the transformation between the two sets of coordinates - the one without the
dislocation and the second with the dislocation )
$e^{a}_{\mu}e^{a}_{\nu}=g_{\mu,\nu}$, defines the Jacobian $\sqrt{G}$ where
$G=det[g_{\mu,\nu}]$. We find that the derivative for a spinor component
$\Psi^{(\alpha)}(\vec{r})$,
$\alpha=[1=1\uparrow;2=1\downarrow;3=2\uparrow;4=2\downarrow]$ is replaced by
the $covariant$ derivative Green :

$\nabla_{\mu}\Psi^{(\alpha)}(\vec{r})=\partial_{\mu}\Psi^{(\alpha)}(\vec{r})+\frac{1}{8}\omega^{(a,b)}_{\mu}[\hat{\Gamma}^{a},\hat{\Gamma}^{b}]^{\alpha}_{\beta}\Psi^{(\beta)}(\vec{r})$
(2)

where $\hat{\Gamma}^{a}$ ,$a=1,2,3,4,5$ are the matrixes:
$\hat{\Gamma}^{1}=-\Gamma^{2}\equiv-(\sigma_{2}\otimes\tau_{1})$;
$\hat{\Gamma}^{2}=\Gamma^{1}\equiv(\sigma_{1}\otimes\tau_{1})$;
$\hat{\Gamma}^{3}=\Gamma^{3}\equiv(\sigma_{3}\otimes\tau_{1})$;
$\hat{\Gamma}^{4}=\Gamma^{4}\equiv(I\otimes\tau_{2})$;$\hat{\Gamma}^{5}=\Gamma^{5}\equiv(I\otimes\tau_{3})$.

The $spin$ $connection$ $\omega^{a,b}_{\mu}$ determines the covariant
derivative Green  is given in terms of the tangent vectors $e^{a}_{\mu}$:
$e^{a}_{\mu}=\partial_{\mu}X^{a}(\vec{r})$; $a=1,2,3$ ; $\mu=x,y,z$.

$\displaystyle\omega^{a,b}_{\mu}=\frac{1}{2}e^{\nu,a}(\partial_{\mu}e^{b}_{\nu}-\partial_{\nu}e^{b}_{\mu})-\frac{1}{2}e^{\nu,b}(\partial_{\mu}e^{a}_{\nu}-\partial_{\nu}e^{a}_{\mu})$
$\displaystyle-\frac{1}{2}e^{\rho,a}e^{\sigma,b}(\partial_{\rho}e_{\sigma,c}-\partial_{\sigma}e_{\rho,c})e^{c}_{\mu}$
(3)

We notice the asymmetry between $e^{\nu,a}$ and $e_{a,\nu}$: $e^{\nu,a}\equiv
g^{\nu,\lambda}e^{a}_{\lambda}$ and $e_{a,\nu}\equiv\delta_{a,b}e^{b}_{\nu}$.
As a result the Hamiltonian in eq.$(1)$ in the second quantized form is
replaced by:

$\displaystyle H^{(3D)}=\hbar
v_{0}\int\,d^{3}r\sqrt{G}[\Psi^{\dagger}(\vec{r})[e^{\mu}_{a}\hat{\Gamma}^{a}(-i\nabla_{\mu})-E_{F}(I\otimes
I)+\hat{\Gamma}^{5}(-M_{0})]\Psi(\vec{r})$
$\displaystyle+B_{1}g^{\mu,\nu}(\nabla_{\mu}\Psi^{\dagger}_{1}(\vec{r}))(\nabla_{\nu}\Psi_{1}(\vec{r}))-B_{1}g^{\mu,\nu}(\nabla_{\mu}\Psi^{\dagger}_{2}(\vec{r})\nabla_{\nu}\Psi_{2})]$

where
$e^{\mu}_{a}\hat{\Gamma}^{a}=\sum_{a}e^{\mu}_{a}\hat{\Gamma}^{a}\equiv\hat{\Gamma}^{\mu}(\vec{r})$,
$[\hat{\Gamma}^{\mu}(\vec{r})\hat{\Gamma}^{\nu}(\vec{r})+\hat{\Gamma}^{\mu}(\vec{r})\hat{\Gamma}^{\nu}(\vec{r})]=2g^{\mu,\nu}(\vec{r})$
, $det[g^{\mu,\nu}(\vec{r})]\equiv G$ and $\nabla_{\mu}$ is the covariant
derivative given in terms of the spin connection given in equation $(2)$:
$\nabla_{\mu}\Psi^{(\alpha)}(\vec{r})=\partial_{\mu}\Psi^{(\alpha)}(\vec{r})+\frac{1}{8}\omega^{(a,b)}_{\mu}[\hat{\Gamma}^{a},\hat{\Gamma}^{b}]^{\alpha}_{\beta}\Psi^{(\beta)}(\vec{r})$

C-The Mechanical strain effect on $H^{(3D)}$

From the work of young  we learn that the effect of the strained field is
different on $Bi_{2}Se_{3}$ than on $Bi_{2}Te_{3}$. In $Bi_{2}Se_{3}$ the
$compressive$ strain decreases the Coulombic gap while increasing the inverted
gap strength induced by the spin-orbit interaction. We will use the result in
equation $(4)$ to analyze the effect of strain. The strain field
$\epsilon_{i,j}$ (symmetric in $i,j$) is related to the stress field
$\sigma_{i,j}$ and elastic stiffness $Lame$ constant $\lambda$ and $\mu$:
$\sigma_{i,j}=\lambda\delta_{i,j}\epsilon_{k,k}+2\mu\epsilon_{i,j}$. Applying
a constant stress $\sigma_{i,j}$ one can determine the value of the constant
strain field $\epsilon_{i,j}$ which is related to the tangent vectors
$e^{i}_{j}\equiv\delta_{i,j}+\epsilon_{i,j}$. In the present case the spin
connection and the Christofel tensor vanish. The metric tensor $g_{i,j}$ is
given by :$g_{i,j}=\delta_{i,j}+2\epsilon_{i,j}$. Using this formulation we
can investigate the effect of the stress on the $Bi_{2}Se_{3}$ at the $\Gamma$
point $\vec{k}=0$. The TI Hamiltonian given in eq.$(4)$
,$M(\vec{k})=-M_{0}+B_{1}k^{2}_{z}+B_{2}(k^{2}_{x}+k^{2}_{y})$ with the
inverted case $M_{0}>0$ Zhangnew  . The Hamiltonian in eq. $(4)$ is replaced
by:

$\displaystyle H^{(3D-strain)}=\hbar
v_{0}\int\,d^{3}r\sqrt{G}[\Psi^{\dagger}(\vec{r})[\hat{\Gamma}^{a}(\delta_{\mu,a}+\epsilon_{\mu,a})(-i\partial_{\mu}))+\hat{\Gamma}^{4}(-M_{0})+B(1-2\epsilon_{\mu,\nu})\partial_{\mu}\hat{\Gamma}^{4}\partial_{\nu}]\Psi(\vec{r})$
$\displaystyle\approx\hbar
v_{0}\int\,d^{3}r\sqrt{G}[\Psi^{\dagger}(\vec{r})[\hat{\Gamma}^{a}[(\delta_{\mu,a}(1+<\epsilon>)(-i\partial_{\mu}))]+\hat{\Gamma}^{4}(-M_{0})+B(1-2<\epsilon>\delta_{\mu,\nu})\partial_{\mu}\hat{\Gamma}^{4}\partial_{\nu}]\Psi(\vec{r})$

In equation $(5)$ we have used the average strain field $<\epsilon>$,
$<\epsilon>\equiv\frac{\epsilon_{1,1}+\epsilon_{2,2}+\epsilon_{3,3}}{3}$. We
replace the spinor field $\Psi(\vec{r})$ by
$\Psi(\vec{r})\sqrt{(1+<\epsilon>)}\equiv\hat{\Psi}(\vec{r})$. As a result we
obtain:

$\displaystyle H^{(3D-strain)}\approx\hbar
v_{0}\int\,d^{3}r\sqrt{G}[\hat{\Psi}^{\dagger}(\vec{r})\hat{\Gamma}^{\mu}(-i\partial_{\mu})+\hat{\Gamma}^{4}\frac{(-M_{0})}{(1+<\epsilon>)}+B\frac{(1-2<\epsilon>)}{1+<\epsilon>}\partial_{\mu}\hat{\Gamma}^{4}\partial_{\mu}]\hat{\Psi}(\vec{r})$

For the compressive case $<\epsilon>$ is negative,
$<\epsilon>\equiv-|<\epsilon>|$ . As a result we observe that the inverted gap
is enhanced
$\frac{|M_{0}|}{(1+<\epsilon>)}=\frac{|M_{0}|}{(1-|<\epsilon>|)}>|M_{0}|$.

In the same way we can show that the Coulomb interaction is reduced: We
introduce the Hubbard Stratonovici field $a_{0}$ to describe the Coulomb
interactions.

$\displaystyle H^{e-e}=\int\,d^{3}r\sqrt{G}[I(-e)\cdot
a_{0}\Psi^{\dagger}(\vec{r})\Psi(\vec{r})+\frac{(1-2\epsilon_{\mu,\nu})}{2}a_{0}\partial_{\mu}\partial_{\nu}a_{0}]$
$\displaystyle\approx\int\,d^{3}r\sqrt{G}[I(-e)\cdot
a_{0}\Psi^{\dagger}(\vec{r})\Psi(\vec{r})+\frac{(1-2<\epsilon>)}{2}a_{0}\partial_{\mu}\partial_{\nu}a_{0}]$

Next we rescale $a_{0}=\frac{A_{0}}{\sqrt{(1-2<\epsilon>)}}$ and obtain:

$H^{e-e}\approx\int\,d^{3}r\sqrt{G}[I\frac{(-e)}{\sqrt{1-2<\epsilon>}}A_{0}\Psi^{\dagger}(\vec{r})\Psi(\vec{r})+A_{0}\partial_{\mu}\partial_{\mu}A_{0}]$
(8)

We observe that for the compressive case the effective charge
$e_{eff.}\equiv\frac{(-e)}{\sqrt{1-2<\epsilon>}}=\frac{(-e)}{\sqrt{1+2|<\epsilon>|}}$
is reduced and therefore the Coulomb gap decreases, while at the same time the
inverted gap increases,
$\frac{|M_{0}|}{(1+<\epsilon>)}=\frac{|M_{0}|}{(1-|<\epsilon>|)}>|M_{0}|$ in
qualitative agreement with young .

III-The chiral metal with an edge dislocation

A-Description of the Chiral model

The low energy Hamiltonian for the bulk $3D$ $TI$ in the $Bi_{2}Se_{3}$ family
was shown to behave on the boundary surface (the $x,y$\- plane) as a two
dimensional chiral metal nature  .

$\displaystyle
H=\int\,d^{2}r\Psi^{\dagger}(\vec{r})[h^{T.I}-\mu]\Psi(\vec{r})]\equiv\hbar
v_{F}\int\,d^{2}r\Psi^{\dagger}(\vec{r})[i\sigma^{1}\partial_{y}-i\sigma^{2}\partial_{x}-\mu]\Psi(\vec{r})$

$h^{T.I}=\hbar v_{F}[i\sigma^{1}\partial_{y}-i\sigma^{2}\partial_{x}]$ is the
chiral Dirac Hamiltonian in the first quantized language. $v_{F}\approx 5\cdot
10^{5}\frac{m}{sec}$ is the Fermi velocity, $\sigma$ is the Pauli matrix
describing the electron spin and $\mu$ is the chemical potential measured
relative to the Dirac $\Gamma$ point. The Hamiltonian for the two dimensional
surface $L\times L$ describes well the excitations smaller than the bulk gap
of the $3D$ $TI$ at $0.3$ $eV$. Moving away from the $\Gamma$ point, the Fermi
velocity becomes momentum dependent; therefore, we will introduce a momentum
cut off $\Lambda$ to restrict the validity of the Dirac model. The chiral
Dirac model in the Bloch representation takes the form: $h=\hbar
v_{F}(\vec{K}\times\vec{\sigma})\cdot\hat{z}\equiv\hbar
v_{F}(-\sigma^{1}k_{y}+\sigma^{2}k_{x})$ The eigen-spinors for this
Hamiltonian are :
$|u(\vec{K})>=[|u_{\uparrow}(\vec{K})>,|u_{\downarrow}(\vec{K})>]^{T}=|\vec{K}>\otimes[1,ie^{i\chi(k_{x},k_{y})}]^{T}$
where $\chi(k_{x},k_{y})=tan^{-1}(\frac{k_{y}}{k_{x}})$ is the spinor phase
and $\epsilon=\hbar v_{F}\sqrt{k^{2}_{x}+k^{2}_{y}}$ is the eigenvalue for
particles . For holes we have the eigenvalue $\epsilon=-\hbar
v_{F}\sqrt{k^{2}_{x}+k^{2}_{y}}$ and eigenvectors
$|v(\vec{K})>=[|v_{\uparrow}(\vec{K})>,|v_{\downarrow}(\vec{K})>]^{T}=|\vec{K}>\otimes[-1,ie^{i\chi(k_{x},k_{y})}]^{T}$.
The chirality operator is defined in terms of the chiral phase
$\chi(k_{x},k_{y})$:

$(\vec{\sigma}\times\frac{\vec{K}}{|\vec{K}|})\cdot\hat{z}\equiv\sin[\chi(k_{x},k_{y})]\sigma^{1}-\cos[\chi(k_{x},k_{y})]\sigma^{2}$
(10)

The chirality operator takes the eigenvalue $-$ (counter-clockwise) for
particles
$[\sin(\chi(k_{x},k_{y}))\sigma^{1}-\cos(\chi(k_{x},k_{y}))\sigma^{2}]|\vec{K}>\otimes[1,ie^{i\chi(k_{x},k_{y})}]^{T}=-|\vec{K}>\otimes[1,ie^{i\chi(k_{x},k_{y})}]^{T}$
and $+$ (clockwise) for holes
$[\sin(\chi(k_{x},k_{y}))\sigma^{1}-\cos(\chi(k_{x},k_{y}))\sigma^{2}]|\vec{K}>\otimes[-1,ie^{i\chi(k_{x},k_{y})}]^{T}=|\vec{K}>\otimes[-1,ie^{i\chi(k_{x},k_{y})}]^{T}$.

B-The effect of edge dislocation on a two dimensional chiral surface
Hamiltonian

We use the notation $x^{\mu}$ ,$\mu=x,y$ and $X^{a}$ ,$a=1,2$ to describe the
media with dislocations. For an edge dislocation in the $x$ direction the
$Burger$ vector $B^{(2)}$ is in the $y$ direction . The value of the burger
vector $B^{(2)}$ is given by the shortest translation lattice vector in the
$y$ direction. (For the $TI$ $Bi_{2}Se_{3}$ the length of the vector $B^{(2)}$
is $5$ times the inter atomic distance ). Following kleinert  we introduce the
coordinate transformation for an edge dislocation:
$\vec{r}=(x,y)\rightarrow[X(\vec{r})=x,Y(\vec{r})=y+\frac{B^{(2)}}{2\pi}\tan^{-1}(\frac{y}{x})]$
with the core of the dislocation centered at $\vec{r}=(0,0)$. The matrix
elements fields $e^{a}_{\mu}$ for the edge dislocation is given by :

$e^{a}_{\mu}=\partial_{\mu}X^{a}(\vec{r});\hskip 7.22743pta=1,2;\hskip
7.22743pt\mu=x,y$ (11)

We express the Burger vector in terms of the the partial derivatives with
respect the coordinates $a=1,2$ in the dislocation frame and $\mu=x,y$ for the
fixed Cartesian frame kleinert :

$\partial_{x}e^{2}_{y}-\partial_{y}e^{2}_{x}=B^{(2)}\delta^{2}(\vec{r})$ (12)

Using Stokes theorem, we replace the line integral $\displaystyle\oint
dx^{\mu}e^{2}_{\mu}(\vec{r})$ by the surface integral $\int\int
dx^{\mu}dx^{\nu}[\partial_{x}e^{2}_{y}-\partial_{y}e^{2}_{x}]$. For a system
with zero $curvature$ and non zero $torsion$ $T^{(2)}_{\mu,\nu}$ we find that
the surface torsion tensor integral $\int\int
dx^{\mu}dx^{\nu}T^{(2)}_{\mu,\nu}$ is equal to $\int\int
dx^{\mu}dx^{\nu}[\partial_{x}e^{2}_{y}-\partial_{y}e^{2}_{x}]$, and therefore
both integrals are equal to the Burger vector.

$\displaystyle\displaystyle\oint dx^{\mu}e^{2}_{\mu}(\vec{r})=\int\int
dx^{\mu}dx^{\nu}[\partial_{\mu}e^{2}_{\nu}-\partial_{\nu}e^{2}_{\mu}]=B^{(2)};$
$\displaystyle\int\int dx^{\mu}dx^{\nu}T^{(2)}_{\mu,\nu}=\int\int
dx^{\mu}dx^{\nu}[\partial_{\mu}e^{2}_{\nu}-\partial_{\nu}e^{2}_{\mu}]=B^{(2)};$

where $dx^{\mu}dx^{\nu}$ represents the surface element. The tangent
components $e^{a}_{\mu}$ can be expressed in terms of the Burger vector
density $B^{(2)}\delta^{2}(\vec{r})$ kleinert  :

$\displaystyle e^{2}_{x}=(\frac{B^{(2)}}{2\pi})\frac{y}{(x^{2}+y^{2})};\hskip
21.68121pte^{2}_{y}=1-(\frac{B^{(2)}}{2\pi})\frac{x}{(x^{2}+y^{2})}$
$\displaystyle e^{1}_{x}=1;\hskip 28.90755pte^{1}_{y}=0$ (14)

Using the tangent components, we obtain the metric tensor $g_{\mu,\nu}$.

$e^{a}_{\mu}e^{a}_{\nu}\equiv
e^{1}_{\mu}e^{1}_{\nu}+e^{2}_{\mu}e^{2}_{\nu}=g_{\mu,\nu}(\vec{r});\hskip
7.22743pte^{a}_{\mu}e^{b}_{\mu}\equiv
e^{a}_{x}e^{b}_{x}+e^{a}_{y}e^{b}_{y}=\delta_{a,b}$ (15)

The inverse of the metric tensor $g_{\mu,\nu}(\vec{r})$ is the tensor
$g^{\nu,\mu}(\vec{r})$ defined trough the equation
$g_{\mu,\tau}(\vec{r})g^{\tau,\nu}(\vec{r})=\delta_{\mu}^{\nu}$. Using the
tangent vectors, we find $to$ $first$ $order$ in the Burger vector the metric
tensor $g_{\mu,\nu}$ and the Jacobian transformation $\sqrt{G}$:

$g_{x,x}=1;\hskip
7.22743ptg_{x,y}=\frac{B^{(2)}}{2\pi}\frac{y}{x^{2}+y^{2}};\hskip
7.22743ptg_{y,y}=1-\frac{B^{(2)}}{2\pi}\frac{y}{x^{2}+y^{2}};\hskip
7.22743ptg_{y,x}=0;\hskip
7.22743ptG=det[g_{\mu,\nu}]=1-\frac{B^{(2)}}{2\pi}\frac{y}{x^{2}+y^{2}}$ (16)

The inverse tensor is given by:$g^{x,x}\approx 1$,
$g^{x,y}=g^{y,x}=-\frac{B^{(2)}}{2\pi}\frac{y}{x^{2}+y^{2}}$,
$g^{y,y}=1+\frac{B^{(2)}}{\pi}\frac{x}{x^{2}+y^{2}}$. Using the inverse tensor
$g^{\mu,\nu}$ we obtain the inverse matrix $e^{\mu}_{a}$ which is given by:

$e^{\mu}_{a}=e_{a,\nu}g^{\nu,\mu}=(\delta_{a,b}e^{b}_{\nu})g^{\nu,\mu}=e^{a}_{\nu}g^{\nu,\mu}$
(17)

Using the components $e^{\mu}_{a}$ we compute the the transformed Pauli
matrices. The Hamiltonian in the absence of the edge dislocation is given by
$h^{T.I.}=i\gamma^{a}\partial_{a}\equiv\sum_{a=1,2}i\gamma^{a}\partial_{a}$
where the Pauli matrices are given by $\gamma^{1}=-\sigma^{2}$ ,
$\gamma_{2}=\sigma_{1}$ and $\gamma^{3}=\sigma^{3}$. (We will use the
convention that when an index appears twice we perform a summation over this
index.) In the presence of the edge dislocation, the term
$\gamma^{a}\partial_{a}$ must be expressed in terms of the Cartesian fixed
coordinates $\mu=x,y$. As a result, the spinor $\Psi(\vec{r})$ transforms
accordingly to the $SU(2)$ transformation . If $\widetilde{\Psi}(\vec{R})$ is
the spinor for the deformed lattice, it can be related with the help of an
$SU(2)$ transformation to the spinor $\Psi(\vec{r})$ in the undeformed
lattice:
$\widetilde{\Psi}(X,Y)=e^{-i\frac{\delta\varphi(x,y)}{2}\sigma^{3}}\Psi(x,y)$
. Where $\delta\varphi(x,y)$ is the rotation angle between the two set of
coordinates: $\delta\varphi(x,y)=tan^{-1}(\frac{Y}{X})-tan^{-1}(\frac{y}{x})$.
Using the relation between the coordinates $X=x$, and
$Y=y+\frac{B^{(2)}}{2\pi}tan^{-1}(\frac{y}{x})$ with the singularity at
$x=y=0$ gives us that the derivative of the phase which is a delta function,
$\partial_{x}\delta\varphi(x,y)=-\partial_{y}\delta\varphi(x,y)\propto\delta^{2}(x,y)$.
Combining the transformation of the derivative with the $SO(2)$ rotation in
the plane, we obtain the form of the chiral Dirac equation in the Cartesian
space (see Appendix A) given in terms of the $spin$ $connection$
$\omega_{\mu}^{1,2}$ Nakahara :

$i\gamma_{a}\partial_{a}\widetilde{\Psi}(\vec{R})=i\delta_{a,b}\gamma^{b}\partial_{a}\widetilde{\Psi}(\vec{R})=i\gamma^{a}e^{\mu}_{a}[\partial_{\mu}+\frac{1}{4}[\gamma^{b},\gamma^{c}]\omega_{\mu}^{bc}]\Psi(\vec{r})$
(18)

The Hamiltonian $h^{T.I.}\rightarrow h^{edge}$ is transformed to the
dislocation edge Hamiltonian with the explicit form given by:

$\displaystyle
h^{edge}=i\sigma^{1}\partial_{2}-i\sigma^{2}\partial_{1}=i\sigma^{1}e^{\mu}_{2}[\partial_{\mu}+\frac{1}{8}[\sigma^{1},\sigma^{2}]\omega^{1,2}_{\mu}]-i\sigma^{2}e^{\mu}_{1}[\partial_{\mu}+\frac{1}{8}[\sigma^{1},\sigma^{2}]\omega^{1,2}_{\mu}]$
$\displaystyle=i(\sigma^{1}e^{\mu}_{2}-\sigma^{2}e^{\mu}_{1})(\partial_{\mu}+\frac{1}{8}[\sigma^{1},\sigma^{2}]\omega^{1,2}_{\mu})$

To first order in the Burger vector we find :
$\omega_{x}^{12}=-\omega_{x}^{21}=0$ and
$-\omega_{y}^{21}=\omega_{y}^{1,2}=-\frac{B^{(2)}}{2}\delta^{2}(\vec{r})$, see
eqs. $(72-74)$ in Appendix A.

$h^{edge}\approx
i\sigma^{1}(\partial_{y}-\frac{i}{2}\sigma^{3}B^{(2)}\delta^{2}(\vec{r}))-i\sigma^{2}\partial_{x}$
(20)

In the second quantized form the chiral Dirac Hamiltonian in the presence of
an edge dislocations is given by :

$\displaystyle
H^{edge}\approx\int\,d^{2}r\sqrt{G}\Psi^{\dagger}(\vec{r})[h^{edge}-\mu]\Psi(\vec{r})$
$\displaystyle\equiv\hbar
v_{F}\int\,d^{2}r\sqrt{G}\Psi^{\dagger}(\vec{r})[i\sigma^{1}(\partial_{y}-\frac{i}{2}\sigma^{3}B^{(2)}\delta^{2}(\vec{r}))-i\sigma^{2}\partial_{x}-\mu]\Psi(\vec{r})$

$h^{edge}$ is the Hamiltonian in the first quantized language, $\mu$ is the
chemical potential and
$\Psi(\vec{r})=[\Psi_{\uparrow}(\vec{r}),\Psi_{\downarrow}(\vec{r})]^{T}$ is
the two component spinor field.

C- The Identification of the physical contours for the edge Hamiltonian
$h^{edge}$

In order to identify the solutions, we will use the complex representation.
The coordinates in the complex representation are given by,
$z=\frac{1}{2}(x+iy)$, $\overline{z}=\frac{1}{2}(x-iy)$,
$\partial_{z}=\partial_{x}-i\partial_{y}$,
$\partial_{\overline{z}}=\partial_{x}+i\partial_{y}$. In this representation
the two dimensional delta function $\delta^{2}(\vec{r})$ is given by
$\delta^{2}(\vec{r})\equiv\frac{1}{\pi}\partial_{z}(\frac{1}{\overline{z}})=\frac{1}{\pi}\partial_{\overline{z}}(\frac{1}{z})$
Conformal ; Nair . We will use the edge Hamiltonian $h^{edge}$ and will
compute the eigenfunctions
$u_{\epsilon}(z,\overline{z})=[U_{\epsilon\uparrow}(z,\overline{z}),U_{\epsilon\downarrow}(z,\overline{z})]^{T}$
and
$v_{\epsilon}(z,\overline{z})=V_{\epsilon\uparrow}(z,\overline{z}),V_{\epsilon\downarrow}(z,\overline{z})]^{T}$.
The eigenvalue equation is given by:

$\displaystyle\epsilon
U_{\epsilon\uparrow}(z,\overline{z})=-[\partial_{z}+(\frac{B^{(2)}}{\sqrt{2}\pi})\partial_{z}(\frac{1}{\overline{z}})]U_{\epsilon\downarrow}(z,\overline{z})$
$\displaystyle\epsilon
U_{\epsilon\downarrow}(z,\overline{z})=[\partial_{\overline{z}}+(\frac{B^{(2)}}{\sqrt{2}\pi})\partial_{\overline{z}}(\frac{1}{z})]U_{\epsilon\uparrow}(z,\overline{z})$

The eigenfunctions $u_{\epsilon}(z,\overline{z})$ and
$v_{\epsilon}(z,\overline{z})$ can be written with the help of a singular
matrix $M(z,\overline{z})$ Ezawa  :

$u_{\epsilon}(z,\overline{z})=M(z,\overline{z})\hat{F}_{\epsilon}(z,\overline{z})\equiv\left[\begin{array}[]{rrr}e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{z})}&0\\\
0&e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{\overline{z}})}\\\
\end{array}\right]\left(\begin{array}[]{cc}F_{\epsilon\uparrow}(z,\overline{z})\\\
F_{\epsilon\downarrow}(z,\overline{z})\end{array}\right)$

($F_{\epsilon}(z,\overline{z})$ and $F_{-\epsilon}(z,\overline{z})$ are the
transformed eigenfunctions for $\epsilon>0$ and $\epsilon<0$ respectively .)
In terms of the transformed spinors the eigenvalue equation
$h^{edge}(z,\overline{z})u_{\epsilon}(z,\overline{z})=\epsilon
u_{\epsilon}(z,\overline{z})$ and $F_{\epsilon\downarrow}(z,\overline{z})$
becomes:

$\epsilon\left(\begin{array}[]{cc}F_{\epsilon\uparrow}(z,\overline{z})\\\
F_{\epsilon\downarrow}(z,\overline{z})\end{array}\right)=\left[\begin{array}[]{rrr}I(z,\overline{z})&0\\\
0&(I(z,\overline{z})^{*}\\\
\end{array}\right]\left[\begin{array}[]{rrr}-\partial_{z}&0\\\
0&\partial_{\overline{z}}\\\
\end{array}\right]\left(\begin{array}[]{cc}F_{\epsilon\uparrow}(z\overline{z})\\\
F_{\epsilon\downarrow},(z,\overline{z})\end{array}\right)$

where
$I(z,\overline{z})=e^{-\frac{B^{(2)}}{2\pi}(\frac{\overline{z}-z}{z\overline{z}})}\equiv
e^{2\frac{B^{(2)}}{\pi}(\frac{iy}{x^{2}+y^{2}})}$ ,
$(I(z,\overline{z}))^{*}=e^{2\frac{B^{(2)}}{\pi}(\frac{-iy}{x^{2}+y^{2}})}$,
$|I(z,\overline{z})|=1$. We search for zero modes $\epsilon=0$ and find :

$\partial_{z}F_{\epsilon\downarrow}(z,\overline{z})=0\hskip
36.135pt\partial_{\overline{z}}F_{\epsilon\uparrow}(z,\overline{z})=0$ (23)

The solutions are given by the holomorphic representation
$F_{\epsilon=0\uparrow}(z,\overline{z})=f_{\uparrow}(z)$ and the anti-
holomorphic function
$F_{\epsilon=0\downarrow}(z,\overline{z})=f_{\downarrow}(\overline{z})$. The
zero mode eigenfunctions are given by :

$u_{\epsilon=0,\uparrow}(z)=e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{z})}f_{\uparrow}(z),\hskip
36.135ptu_{\epsilon=0,\downarrow}(\overline{z})=e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{\overline{z}})}f_{\downarrow}(\overline{z})$
(24)

Due to the presence of the essential singularity at $z=0$ it is not possible
to find analytic functions $f_{\uparrow}(z)$ and
$f_{\downarrow}(\overline{z})$ which vanish fast enough around $z=0$ such that
$\int d^{2}z(u_{\epsilon=0,\lambda}(z))^{*}u_{\epsilon=0,\lambda}(z)<\infty$.
Therefore, we conclude that zero mode solution does not exists. The only way
to remedy the problem is to allow for states with finite energy.

In the next step we look for finite energy states. We perform a coordinate
transformation :

$z\rightarrow W[z,\overline{z}];\hskip
36.135pt\overline{z}\rightarrow\overline{W}[z,\overline{z}]$ (25)

We demand that the transformation is conformal and preserve the orientation.
This restricts the transformations to holomorphic and anti-holomorphic
functions Conformal . This means that we have the conditions
$\partial_{\overline{z}}W[z,\overline{z}]=0$ and
$\partial_{z}\overline{W}[z,\overline{z}]=0$. As a result we obtain
$W[z,\overline{z}]=W[z]$ and
$\overline{W}[z,\overline{z}]=\overline{W}[\overline{z}]$, which obey the
eigenvalue equations:

$\displaystyle\epsilon
F_{\epsilon\uparrow}(W,\overline{W})=-\partial_{W}F_{\epsilon\downarrow}(W,\overline{W})$
$\displaystyle\epsilon
F_{\epsilon\downarrow}(W,\overline{W})=\partial_{\overline{W}}F_{\epsilon\uparrow}(W,\overline{W})$

This implies the conditions $\frac{dW[z]}{dz}=(I(z,\overline{z}))^{*}$ and
$\frac{d\overline{W}[\overline{z}]}{dz}=I(z,\overline{z})$. Since
$I(z,\overline{z})$ is neither holomorphic or anti-holomorphic and satisfy
$|I(z,\overline{z})|=1$, the only solutions for $W[z]$ and
$\overline{W}[\overline{z}]$ must obey $I(z,\overline{z})=1$:

$I(z,\overline{z})\equiv
e^{2\frac{B^{(2)}}{\pi}(\frac{iy}{x^{2}+y^{2}})}=e^{i2\pi n};\hskip
14.45377ptn=0,\pm 1,\pm 2....$ (27)

For $I(z,\overline{z})\neq 1$ one obtains solutions which are unstable . The
stable solutions will be given by a one parameter $s$ curve ($s$ is the length
of the curve) $\vec{r}(s)\equiv[x(s),y(s)]$ which obey the equation
$I(z,\overline{z})=1$. The curve $\vec{r}(s)$ allows us to define the
$tangent$ $\vec{t}(s)$ and the $normal$ vectors $\vec{N}(s)$. This allows us
to introduce a two- dimensional region in the vicinity of the contour of
$\vec{r}(s)\rightarrow\vec{R}(s,u)=\vec{r}(s)+u\vec{N}(s)$.

IID- The wave function for the edge dislocation-the $n=0$ contour

The condition
$I(z,\overline{z})=e^{2\frac{B^{(2)}}{\pi}(\frac{iy}{x^{2}+y^{2}})}=1$ for
$n=0$ is satisfied for $y=0$ and large value of $y$ which obey
$2\frac{B^{(2)}}{\pi}(\frac{y}{x^{2}+y^{2}})<<1$ . The values of $y$ which
satisfy this conditions are restricted to
$I(z,\overline{z})=e^{2\frac{B^{(2)}}{\pi}(\frac{iy}{x^{2}+y^{2}})}\approx 1$.
This condition is satisfied for values of $y$ in the range:

$2\frac{B^{(2)}}{\pi}(\frac{y}{x^{2}+y^{2}})\leq\eta<\frac{\pi}{4}<1$ (28)

We introduce the radius $R_{g}=\frac{B^{2}}{2\pi^{2}}$ and find that the
condition $I(z,\overline{z})\approx 1$ gives rise to the equation for $y$. The
solution is given by
$x^{2}+(y\pm\frac{2\pi}{\eta}R_{g})^{2}=(\frac{2\pi}{\eta}R_{g})^{2}$.
Therefore, for $|y|>|d|\geq(\frac{2\pi}{\eta})2R_{g}>2R_{g}$ we have $I\approx
1$ which corresponds to a free particle eigenvalue equations.

$\displaystyle\epsilon
F_{\epsilon\uparrow}(x,y)=e^{\frac{B^{(2)}}{\pi}\frac{i2y}{(x^{2}+y^{2})}}[-\partial_{x}+i\partial_{y}]F_{\epsilon\downarrow}(x,y)$
$\displaystyle\approx[-\partial_{x}+i\partial_{y}]F_{\epsilon\downarrow}(x,y);$
$\displaystyle\epsilon
F_{\epsilon\downarrow}(x,y)=e^{\frac{B^{(2)}}{\pi}\frac{-i2y}{(x^{2}+y^{2})}}[\partial_{x}+i\partial_{y}]F_{\epsilon\uparrow}(x,y)$
$\displaystyle\approx[\partial_{x}+i\partial_{y}]F_{\epsilon\uparrow}(x,y)$

For $|y|>d$ the eigenfunctions are given by:
$U_{\epsilon,\uparrow}(x,y)=e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x+iy})}F_{\epsilon,\uparrow}(x,y)$,
$U_{\epsilon,\downarrow}(x,y)=e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x-iy})}F_{\epsilon,\downarrow}(x,y)$
where $F_{\epsilon\uparrow}(x,y)$ and $F_{\epsilon\downarrow}(x,y)$ are the
eigenfunctions of equation (21). The envelope functions
$e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x+iy})}$,
$e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x-iy})}$ which multiply the wave functions
$F_{\epsilon\uparrow}(x,y)$ , $F_{\epsilon\downarrow}(x,y)$ impose vanishing
boundary conditions for the eigenfunctions $U_{\epsilon,\downarrow}(x,y)$ and
$U_{\epsilon,\uparrow}(x,y)$ at $y\rightarrow\pm\infty$ . therefore, we demand
that the eigenfunctions $U_{\epsilon,\uparrow}(x,y)$,
$U_{\epsilon,\downarrow}(x,y)$ should vanish at the boundaries
$y=\pm\frac{L}{2}$. Since the multiplicative envelope functions for opposite
spins is complex conjugate to each other we have to make the choice that one
of the spin components vanishes at one side and the other component at the
opposite side. Two possible choices can be made:

$U_{\epsilon,\uparrow}(x,y=\frac{L}{2})\equiv
e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x+i\frac{L}{2}})}F_{\epsilon\uparrow}(x,\frac{L}{2})=U_{\epsilon,\downarrow}(x,y=-\frac{L}{2})\equiv
e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x-i(-\frac{L}{2})})}F_{\downarrow}(x,-\frac{L}{2})=0$

or

$U_{\epsilon,\uparrow}(x,y=-\frac{L}{2})\equiv
e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x+i(-\frac{L}{2})})}F_{\epsilon\uparrow}(x,-\frac{L}{2})=U_{\epsilon,\downarrow}(x,y=\frac{L}{2})\equiv
e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x-i\frac{L}{2}})}F_{\downarrow}(x,\frac{L}{2})=0$

Making the first choice, (both choices give the same eigenvalues and
eigenfunction) we compute the eigenfunctions $F_{\epsilon\uparrow}(x,y)$ and
$F_{\epsilon\downarrow}(x,y)$ for $|y|>d$ using the boundary conditions :

$F_{\epsilon\uparrow}(x,y=\frac{L}{2})=0;\hskip
28.90755ptF_{\epsilon\downarrow}(x,y=-\frac{L}{2})=0$ (30)

Due to the fact that the solutions are restricted to $|y|>d$ no conditions
need to be imposed at $x=y=0$. In the present case we consider a situation
with a single dislocation. This is justified for a dilute concentration of
dislocations typically separated by a distance $l\approx 10^{-6}m$. ( In
principle we need at least two dislocations in order to satisfy the condition
that the sum of the Burger vectors is zero.) The eigenvalues are given by
$\epsilon=\pm\hbar v_{F}\sqrt{p^{2}+q^{2}}$. The value of $p$ is determined by
the periodic boundary condition in the $x$ direction
$p(m)=\frac{2\pi}{L}m\equiv\frac{2\pi}{Na}m$, $m=0,1,...,(N-2),(N-1)$ and $a$
is the lattice constant $a\approx\frac{2\pi}{\Lambda}$. The value of $q$ will
be obtained from the vanishing boundary conditions at $y=\pm\frac{L}{2}$. The
eigenfunctions $F_{\epsilon,\sigma}(x,y)$ will be obtained using the linear
combination of the spinors introduced in chapter $III$. In the Cartesian
representation we can build four spinors $\Gamma_{p,q}(x,y)$,
$\Gamma_{p,-q}(x,y)$,$\Gamma_{-p,q}(x,y)$,$\Gamma_{-p,-q}(x,y)$ which are
eigenstates of the chirality operator and are given by:

$\Gamma_{p,q}(x,y)=e^{ipx}e^{iqy}\left(\begin{array}[]{cc}1\\\
ie^{i\chi(p,q)}\end{array}\right)$
$\Gamma_{p,-q}(x,y)=e^{ipx}e^{-iqy}\left(\begin{array}[]{cc}1\\\
ie^{-i\chi(p,q)}\end{array}\right)$
$\Gamma_{-p,q}(x,y)=e^{ipx}e^{iqy}\left(\begin{array}[]{cc}1\\\
-ie^{-i\chi(p,q)}\end{array}\right)$
$\Gamma_{-p,-q}(x,y)=e^{-ipx}e^{-iqy}\left(\begin{array}[]{cc}1\\\
-ie^{i\chi(p,q)}\end{array}\right)$ (31)

where $tan[\chi(p,q)]=\frac{q}{p}$.

The $TI$ Hamiltonian $h^{T.I}(x,y)=\hbar
v_{F}[i\sigma^{1}\partial_{y}-i\sigma^{2}\partial_{x}]$ is invariant under the
symmetry the operation $x\rightarrow-x$ which is described by the
transformation $P_{x}$ : $P_{x}xP^{-1}_{x}=-x$;
$P_{x}\sigma^{2}P^{-1}_{x}=-\sigma^{2}$; $P_{x}yP^{-1}_{x}=y$;
$P_{x}\sigma^{1}P^{-1}_{x}=\sigma^{1}$.

The edge Hamiltonian $h^{edge}$ contains in addition the term
$\sigma^{2}\delta(\vec{r})$ which changes sign under the symmetry operation
$P_{x}$ . As a result the symmetry operation does not commute with the edge
Hamiltonian $[h^{edge},P_{x}]\neq 0$. This result demands that we construct
two independent eigenfunctions $F^{(n=0,R)}_{p>0,q}(x,y)$ ($right-mover$) for
$p>0$ and $F^{(n=0,L)}_{-p>0,q}(x,y)$ ($left-mover$) $p<0$.

$\displaystyle
F^{(n=0,R)}_{p>0,q}(x,y)=A(q)\Gamma_{p,q}(x,y)+B(q)\Gamma_{p,-q}(x,y)$
$\displaystyle
F^{(n=0,L)}_{-p>0,q}(x,y)=C(q)\Gamma_{-p,q}(x,y)+D(q)\Gamma_{-p,-q}(x,y)$

Employing the boundary conditions given in equation $(29)$ we obtain the
amplitudes $\frac{D(q)}{C(q)}$ , $\frac{B(q)}{A(q)}$ and the discrete momenta
$q_{+}$. Using the pair $\Gamma_{p,q}(x,y)$ , $\Gamma_{p,-q}(x,y)$ $p>0$ we
obtain :

$\displaystyle
F^{(n=0,R)}_{\epsilon(p>0,q_{+}),\uparrow}(x,y)=e^{ipx}e^{\frac{i}{2}\chi(p,q_{+})}[e^{i(q_{+}y-\frac{1}{2}\chi(p,q_{+}))}+(-1)^{k+1}e^{-i(q_{+}y-\frac{1}{2}\chi(p,q_{+}))}];|y|>d$
$\displaystyle
F^{(n=0,R)}_{\epsilon(p>0,q_{+}),\downarrow}(x,y)=ie^{ipx}e^{\frac{i}{2}\chi(p,q_{+})}[e^{i(q_{+}y+\frac{1}{2}\chi(p,q_{+}))}+(-1)^{k+1}e^{-i(q_{+}y+\frac{1}{2}\chi(p,q_{+}))}];|y|>d$
$\displaystyle q\equiv
q_{+}=\frac{\pi}{L}k+\frac{1}{L}\tan^{-1}(\frac{q_{+}}{p});k=1,2,3...;\tan[\chi(p,q_{+})]=(\frac{q_{+}}{p})$
$\displaystyle\epsilon(p,q_{+})=\pm\hbar
v_{F}\sqrt{(\frac{2\pi}{L}m)^{2}+q_{+}^{2}}$

Similarly, for the second pair $\Gamma_{-p,q}(x,y)$,$\Gamma_{-p,-q}(x,y)$,
$p>0$ we obtain:

$\displaystyle
F^{(n=0,L)}_{\epsilon(-p>0,q_{-}),\uparrow}(x,y)=e^{-ipx}e^{-\frac{i}{2}\chi(p,q_{-})}[e^{i(q_{-}y+\frac{1}{2}\chi(p,q_{-}))}+(-1)^{k+1}e^{-i(q_{-}y+\frac{1}{2}\chi(p,q_{-}))}];|y|>d$
$\displaystyle
F^{(n=0,L)}_{\epsilon(-p>0,q_{-}),\downarrow}(x,y)=-ie^{-ipx}e^{-\frac{i}{2}\chi(p,q_{-})}[e^{i(q_{-}y-\frac{1}{2}\chi(p,q_{-}))}+(-1)^{k+1}e^{-i(q_{-}y-\frac{1}{2}\chi(p,q_{-}))}];|y|>d$
$\displaystyle q\equiv
q_{-}=\frac{\pi}{L}k-\frac{1}{L}\tan^{-1}(\frac{q_{-}}{p});k=1,2,3...;\tan[\chi(p,q_{-})]=(\frac{q_{-}}{p})$
$\displaystyle\epsilon(-p,q_{-})=\pm\hbar
v_{F}\sqrt{(\frac{2\pi}{L}m)^{2}+q_{-}^{2}}$

For the state with zero momentum $p=0$ we find:

$\displaystyle
F^{(n=0,0)}_{\epsilon(p=0,q),\uparrow}(x,y)=2e^{\frac{-i\pi}{4}}\cos[qy+\frac{\pi}{4}];|y|>d$
$\displaystyle
F^{(n=0,0)}_{\epsilon(p=0,q),\downarrow}(x,y)=i2e^{\frac{-i\pi}{4}}\cos[qy-\frac{\pi}{4}];|y|>d$
$\displaystyle q=\frac{\pi}{2L}+\frac{\pi}{L}k;k=0,1,2,3...$
$\displaystyle\epsilon(p=0,q)=\pm\hbar v_{F}|q|$

The eigenfunctions for the dislocation problem for $|y|>d$ will be given in
terms of the envelope functions $e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x+iy})}$ ,
$e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x-iy})}$
($U_{\epsilon,\uparrow}(x,y)=e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x+iy})}F_{\epsilon,\uparrow}(x,y)$,
$U_{\epsilon,\downarrow}(x,y)=e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x-iy})}F_{\epsilon,\downarrow}(x,y)$).

The explicit solutions are given by :

$u^{(n=0,R)}_{\epsilon}(x,y)\equiv[U^{(n=0,R}_{\epsilon\uparrow}(x,y),U^{(n=0,R)}_{\epsilon\downarrow}(x,y)]^{T}$;
$u^{(n=0,L)}_{\epsilon}(x,y)\equiv[U^{(n=0,L}_{\epsilon\uparrow}(x,y),U^{(n=0,L)}_{\epsilon\downarrow}(x,y)]^{T}$.

The components of the spinor are given by:

$\displaystyle
U^{(n=0,R)}_{\uparrow}(x,y)\approx\frac{2const.(B^{(2)})}{G^{\frac{1}{4}}(x,y)L}e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x+iy})}F^{(n=0,R)}_{\epsilon(p>0,q_{+}),\uparrow}(x,y)$
$\displaystyle
U^{(n=0,R)}_{\downarrow}(x,y)\approx\frac{2const.(B^{(2)})}{G^{\frac{1}{4}}(x,y)L}e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x-iy})}F^{(n=0,R)}_{\epsilon(p>0,q_{+}),\downarrow}(x,y)$
$\displaystyle
U^{(n=0,L)}_{\uparrow}(x,y)\approx\frac{2const.(B^{(2)})}{G^{\frac{1}{4}}(x,y)L}e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x+iy})}F^{(n=0,L)}_{\epsilon(-p>0,q_{-}),\uparrow}(x,y)$
$\displaystyle
U^{(n=0,L)}_{\downarrow}(x,y)\approx\frac{2const.(B^{(2)})}{G^{\frac{1}{4}}(x,y)L}e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x-iy})}F^{(n=0,L)}_{\epsilon(-p>0,q_{-}),\downarrow}(x,y)$
$\displaystyle U^{(n=0,0)}_{\uparrow}(x,y)\approx
e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x+iy})}F^{(n=0,0)}_{\epsilon(p=0,q),\uparrow}(x,y)$
$\displaystyle U^{(n=0,0)}_{\downarrow}(x,y)\approx
e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x-iy})}F^{(n=0,0)}_{\epsilon(p=0,q),\downarrow}(x,y)$

where $G(x,y)=1-\frac{B^{(}2)}{2\pi}\frac{y}{\sqrt{2}(x^{2}+y^{2})}$ is the
Jacobian introduced by the edge dislocation. The eigenstates are normalized
and
obey:$\int\,dx\int\,dy\sqrt{G(x,y)}(U^{(n=0,R)}_{\sigma}(x,y))^{*}U^{(n=0,R)}_{\sigma^{\prime}}(x,y)\approx\delta_{\sigma,\sigma^{\prime}}$,
$\int\,dx\int\,dy\sqrt{G(x,y)}(U^{(n=0,L)}_{\sigma}(x,y))^{*}U^{(n=0,L)}_{\sigma^{\prime}}(x,y)\approx\delta_{\sigma,\sigma^{\prime}}$.
The normalization factor $\frac{2const.(B^{(2)})}{L}\approx\frac{2}{L}$, has a
weak dependence on the Burger vector $B^{(2)}$ . This dependence is a
consequence of the Jacobian $\sqrt{G}$ which affects the normalization
constant.

(The multiplicative factor $e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x\pm iy})}$
gives rise to a weak non-orthogonality between the states. This non-
orthogonality of the linear independent eigenfunctions can be corrected with
the help of the Grahm-Shmidt method.)

For the present case, backscattering is allowed but it is much weaker in
comparison to regular metals. This is seen as follows: Time reversal is not
violated; due to the parity violation, the eigenstates
$u^{(n=0,R)}_{\epsilon}(x,y)$ ,$u^{(n=0,L)}_{\epsilon}(x,y)$ are not related
by a time reversal symmetry ($Tu^{(n=0,R)}_{\epsilon}(x,y)\neq
u^{(n=0,L)}_{\epsilon}(x,y)$) . As a result, the backscattering potential
$V_{p,-p}$ is controlled by a finite matrix element between states with
different eigenvalues $\epsilon(-p,q_{-})\neq\epsilon(p,q_{+})$ (contrary to
regular metals where the impurity potential $V_{p,-p}$ connects states with
the same energy). In the present case
$|\epsilon(-p,q_{-})-\epsilon(p,q_{+})|=\hbar
v_{F}|[\sqrt{(\frac{2\pi}{L}m)^{2}+q_{-}^{2}}-\sqrt{(\frac{2\pi}{L}m)^{2}+q_{+}^{2}}]|\neq
0$ the eigenvalues are not equal, therefore the finite matrix element
controlled by the backscattering potential $V_{p,-p}$ gives rise only to a
second order backscattering effect!

IIIE- The circular contours-the wave function for $n\neq 0$

The equation
$I(z,\overline{z})=e^{2\frac{B^{(2)}}{\pi}(\frac{iy(s)}{x^{2}(s)+y^{2}(s)})}=e^{i2\pi
n}$ gives the set of ring contours for $n=\pm 1,\pm 2,\pm 3,...$ shown in
figure 1. The radius $R_{g}$ for the fundamental contour($n=1$) is represented
in terms of the Burger vector $B^{(2)}$, $R_{g}=\frac{B^{(2)}}{2\pi^{2}}$ and
$R_{g}(n)=\frac{R_{g}}{|n|}$.

$(x(s))^{2}+(y(s)\pm R_{g}(n))^{2}=(R_{g}(n))^{2}$ (37)

The centers of the contours are given by :$[\bar{x},\bar{y}]=[0,R_{g}(n)]$ for
$n\neq 0$. When $n>0$ the center of the contours has positive coordinates
(upper contour) and for $n<0$ the center has negative coordinates (lower
contour). Each contour is characterized by a circle with a radius
$R_{g}(n)\equiv\frac{R_{g}}{|n|}$ centered at $[\bar{x}=0,\bar{y}=R_{g}(n)]$.
The contour is parametrized in terms of the arc length $0\leq
s<2\pi\frac{R_{g}}{|n|}$ which is equivalent to $0\leq\varphi<2\pi$ . Each
contour is parametrized by $\vec{r}(s)\equiv[x(s),R_{g}(n)+y(s)]$ where
$x(s)=R_{g}(n)\cos[\frac{s}{R_{g}(n)}]\equiv\ R_{g}(n)\cos[\varphi]$ and
$y(s)=R_{g}(n)\sin[\frac{s}{R_{g}(n)}]\equiv\ R_{g}(n)\sin[\varphi]$. We will
extend this curve to a two dimensional strip with the coordinate $u$ in the
normal direction: For the curve curve $\vec{r}(s)=[x(s),y(s)]$ we will use the
tangent $\vec{t}(s)$ and the normal vector $\vec{N}(s)$ Therefore, the two
dimensional region in the vicinity of the one parameter curve $\vec{r}(s)$ is
replaced by $\vec{r}(s)\rightarrow\vec{R}(s,u)=\vec{r}(s)+u\vec{N}(s)$.

$\displaystyle
x(s,u)=R_{g}(n)\cos[\frac{s}{R_{g}(n)}]+u\cos[\frac{s}{R_{g}(n)}]$
$\displaystyle
y(s,u)=R_{g}(n)\sin[\frac{s}{R_{g}(n)}]+u\sin[\frac{s}{R_{g}(n)}]$

We will restrict the width $|u|$ such that $e^{i2\pi n}e^{\pm i\eta}\approx 1$
where $\eta$ obeys $\eta<\frac{\pi}{4}<1$ ,
$|u|\leq\frac{R_{g}(n)}{1-\frac{\eta}{2\pi n}}-R_{g}(n)\approx
R_{g}(n)(\frac{\eta}{2\pi n})<\frac{R_{g}(n)}{8n}$. In these new coordinates,
the Dirac equation is approximated for $|u|\leq R_{g}(n)(\frac{\eta}{2\pi
n})=\frac{D(n)}{2}$ by :

$\displaystyle\epsilon
F_{\epsilon\uparrow}(s,u)=-I(s,u)e^{-i\frac{s}{R_{g}(n)}}[\partial_{u}-\frac{i}{1+\frac{u}{R_{g}(n)}}\partial_{s}]F_{\epsilon\downarrow}(s,u)\approx-e^{-i\frac{s}{R_{g}(n)}}[\partial_{u}-i\partial_{s}]F_{\epsilon\downarrow}(s,u)$
$\displaystyle\epsilon
F_{\epsilon\downarrow}(s,u)=(I(s,u))^{*}e^{i\frac{s}{R_{g}(n)}}[\partial_{u}+\frac{i}{1+\frac{u}{R_{g}(n)}}\partial_{s}]F_{\epsilon\uparrow}(s,u)\approx
e^{i\frac{s}{R_{g}(n)}}[\partial_{u}+i\partial_{s}]F_{\epsilon\uparrow}(s,u)$

The solution for the contour $n\neq 0$, $0\leq s<2\pi R_{g}(n)$;
$|u|\leq\frac{D(n)}{2}$

The periodicity in $s$ allows us to represent the eigenfunctions in the form:
$F_{\epsilon\uparrow}(s,u)=\sum_{j=-\infty}^{\infty}\sum_{q}e^{ij(\frac{s}{R_{g}(n)})}e^{iqu}F_{\epsilon\uparrow}(j,q)$
and
$F_{\epsilon\downarrow}(s,u)=\sum_{j=-\infty}^{\infty}\sum_{q}e^{i(j+1)(\frac{s}{R_{g}(n)})}e^{iqu}F_{\epsilon\downarrow}(j,q)$.
We find:

$\displaystyle\epsilon
F_{\uparrow}(\epsilon;j,q)=(iq+\frac{j}{R_{g}(n)})F_{\downarrow}(\epsilon;j,q)$
$\displaystyle\epsilon
F_{\downarrow}(\epsilon;j,q)=(iq+\frac{j+1}{R_{g}(n)})F_{\uparrow}(\epsilon;j,q)$

The determinant of the two equations determines the relation between the
eigenvalue $\epsilon$, the transverse momentum $Q(\epsilon)$ and the
eigenfunctions $F_{\epsilon\downarrow}(j,q)$,$F_{\epsilon\uparrow}(j,q)$. The
eigenvalues are degenerate and obey : $\epsilon(j=l;k)=\epsilon(j=-(l+1);k)$
,where $l\geq 0$.

$\displaystyle q\equiv\frac{-i}{2R_{g}(n)}\pm Q(\epsilon);\hskip
14.45377ptQ(\epsilon)=\sqrt{\epsilon^{2}-(\frac{l+\frac{1}{2}}{R_{g}(n)})^{2}}$
$\displaystyle
F_{\epsilon}(l,q)\equiv[F_{\epsilon\uparrow}(l,q),F_{\epsilon\downarrow}(l,q)]^{T}\propto[1,e^{-i\kappa(Q,l)}]^{T};\hskip
7.22743pt\kappa(Q,l)=tan^{-1}(\frac{QR_{g}(n)}{l+\frac{1}{2}})$

The value of the transversal momentum $Q(\epsilon)$ will be determined from
the boundary conditions at $\pm\frac{D(n)}{2}$. We will introduce a polar
angle $\theta$ measured with respect the Cartesian axes: The angle
$0<\varphi(n=1)\leq 2\pi$ for the upper contour $n=1$ centered at
$[\overline{x}=0,\overline{y}=R_{g}]$ is described by the polar coordinate
$0<\theta\leq\pi$ measured from the center of the Cartesian coordinate
$[0,0]$. The lower contour centered at $[\overline{x}=0,\overline{y}=-R_{g}]$
characterized by the angle $0<\varphi(n=-1)\leq 2\pi$ is described by the
polar angle $\theta$ restricted to $\pi<\theta\leq 2\pi$. We establish the
correspondence between $\varphi(n=\pm 1)$ and $\theta$:

$\displaystyle\varphi(n=1)=2\theta+\frac{3\pi}{2}\hskip 7.22743ptfor\hskip
7.22743ptthe\hskip 7.22743ptupper\hskip 7.22743ptcontour\hskip
7.22743ptn=1,\hskip 7.22743pt0<\theta\leq\pi$
$\displaystyle\varphi(n=-1)=2\theta+\frac{3\pi}{2}+\pi\hskip
7.22743ptfor\hskip 7.22743ptthe\hskip 7.22743ptlower\hskip
7.22743ptcontour\hskip 7.22743ptn=-1,\hskip 7.22743pt0<\theta\leq\pi$

Following the discussion from the previous chapter we will introduce the
following boundary conditions:

$\displaystyle F^{(n=1)}_{\epsilon\uparrow}(s,u=\frac{D}{2})=0;\hskip
28.90755ptF^{(n=1)}_{\epsilon\downarrow}(s,y=-\frac{D}{2})=0$ $\displaystyle
F^{(n=-1)}_{\epsilon\uparrow}(s,u=-\frac{D}{2})=0;\hskip
28.90755ptF^{(n=-1)}_{\epsilon\downarrow}(s,y=\frac{D}{2})=0$ $\displaystyle
D(n=\pm 1)\equiv D$ (43)

For the two contours $n=\pm 1$ we introduce eight spinors $\Gamma^{(n=\pm
1)}_{l,Q}(\varphi(n=\pm 1),u)$,$\Gamma^{(n=\pm 1)}_{l,-Q}(\varphi(n=\pm
1),u)$, $\Gamma^{(n=\pm 1)}_{-l,Q}(\varphi(n=\pm 1),u)$, $\Gamma^{(n=\pm
1)}_{-l,-Q}(\varphi(n=\pm 1),u)$. Using this spinor we will compute the
eigenfunctions. For the case $n=0$ we had only four spinors given in equation
$(30)$. The four spinors have been used to construct the eigenfunctions
$F^{(n=0,R)}_{p>0,q}(x,y)$ for $p>0$ and $F^{(n=0,L)}_{-p>0,q}(x,y)$ . Due to
the fact that for each $n\neq 0$ we have two contours $n=\pm$ we have eight
spinors which will be used to construct the eigenfunctions.

$\Gamma^{(n=\pm 1)}_{l,Q}(\varphi(n=\pm 1),u)=e^{il(\varphi(n=\pm
1))}e^{iQu}\left(\begin{array}[]{cc}1\\\ e^{i(\varphi(n=\pm
1))}e^{-i\kappa(l,Q)}\end{array}\right)$ $\Gamma^{(n=\pm
1)}_{l,-Q}(\varphi(n=\pm 1),u)=e^{il(\varphi(n=\pm
1))}e^{-iQu}\left(\begin{array}[]{cc}1\\\ e^{i(\varphi(n=\pm
1))}e^{i\kappa(l,Q)}\end{array}\right)$ $\Gamma^{(n=\pm
1)}_{-l,Q}(\varphi(n=\pm 1),u)=e^{-il(\varphi(n=\pm
1))}e^{iQu}\left(\begin{array}[]{cc}1\\\ -e^{-i(\varphi(n=\pm
1))}e^{i\kappa(l,Q)}\end{array}\right)$ $\Gamma^{(n=\pm
1)}_{-l,-Q}(\varphi(n=\pm 1),u)=e^{-il(\varphi(n=\pm
1))}e^{-iQu}\left(\begin{array}[]{cc}1\\\ -e^{-i(\varphi(n=\pm
1))}e^{-i\kappa(l,Q)}\end{array}\right)$ (44)

Using the vanishing boundary condition given in equation $(42)$ we construct
for this case similar spinors as the one given in equation $(31)$. In the
present case we have for each $n\neq 0$ two contours, therefore the number of
spinors will be doubled. We find instead of the eigenfunction given in
equation $(33)$ two sets of eigenfunctions with momentum $Q_{-}$ (which
replaces $q_{-}$ , see $(33)$) and $Q_{+}$ (which replaces $q_{+}$ , see
$(32)$) .

Using the boundary conditions given in eq.$(35)$ we determine the quantization
conditions $Q_{-}$,$Q_{+}$ and the eigenfunctions for the $n=1$ and $n=-1$
contours.

$\displaystyle
Q_{-}=\frac{\pi}{D}k-\frac{1}{D}\tan^{-1}(\frac{Q_{-}R_{g}(1)}{l+\frac{1}{2}}),k=1,2,3...;\tan[\kappa(l,Q_{-})]=(\frac{Q_{-}R_{g}(1)}{l+\frac{1}{2}})$
$\displaystyle\epsilon(l,Q_{-})=\pm\hbar
v_{F}\sqrt{(\frac{l+\frac{1}{2}}{R_{g}(1)})^{2}+Q_{-}^{2}}$ $\displaystyle
Q_{+}=\frac{\pi}{D}k+\frac{1}{D}\tan^{-1}(\frac{Q_{+}R_{g}(1)}{l+\frac{1}{2}}),k=1,2,3...$
$\displaystyle\tan[\kappa(l,Q_{+})]=(\frac{Q_{+}R_{g}(1)}{l+\frac{1}{2}})$
$\displaystyle\epsilon(l,Q_{+})=\pm\hbar
v_{F}\sqrt{(\frac{l+\frac{1}{2}}{R_{g}(n)})^{2}+Q_{+}^{2}}$ (45)

Using the fact that the combined wave function on the contours $n=1$ and
$n=\pm 1$ must be finite we obtain two sets of wave functions. We include the
envelope function and obtain the wave function for $Q_{-}$ and $Q_{+}$: The
envelope functions $e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x+iy})}$ ,
$e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x-iy})}$ when projected to the contours
take a complicated form. The envelope functions can be expressed in terms of
the functions $\eta(u)$ and $\zeta(\theta,u)$:

$\displaystyle\eta(u)=\frac{R_{g}(1)}{R_{g}(1)+u},\frac{|u|}{R_{g}(1)}<1$
$\displaystyle\zeta(\theta,u)=\frac{-B^{(2)}}{2\pi(R_{g}(1)+u)((\sin[2\theta])^{2}+(\eta(u)-\cos[2\theta])^{2})}$

We find for $Q_{-}$:

$\displaystyle
U_{\epsilon(l,Q_{-})\uparrow}(\theta,u)=G^{\frac{-1}{4}}(\theta,u)\cdot[U^{(even,k)}_{\epsilon(l,Q_{-})\uparrow}(\theta,u)+U^{(odd,k)}_{\epsilon(l,Q_{-})\uparrow}(\theta,u)];$
$\displaystyle
U^{(even,k)}_{\epsilon(l,Q_{-})\uparrow}(\theta,u)=2ie^{\frac{-i}{2}\kappa(l,Q_{-})}[e^{\zeta(\theta,u)\sin[2\theta]}e^{-i\zeta(\theta,u)(\eta(u)-\cos[2\theta])}e^{il(2\theta+\frac{3\pi}{2})}\sin[Q_{-}u+\frac{1}{2}\kappa(l,Q_{-})]$
$\displaystyle+(-1)^{l}e^{-\zeta(\theta,u)\sin[2\theta]}e^{-i\zeta(\theta,u)(-\eta(u)+\cos[2\theta])}e^{-il(2\theta+\frac{3\pi}{2})}\sin[Q_{-}u-\frac{1}{2}\kappa(l,Q_{-})]];$
$\displaystyle
U^{(odd,k)}_{\epsilon(l,Q_{-})\uparrow}(\theta,u)=2e^{\frac{-i}{2}\kappa(l,Q_{-})}[e^{\zeta(\theta,u)\sin[2\theta]}e^{-i\zeta(\theta,u)(\eta(u)-\cos[2\theta])}e^{il(2\theta+\frac{3\pi}{2})}\cos[Q_{-}u+\frac{1}{2}\kappa(l,Q_{-})]$
$\displaystyle+(-1)^{l}e^{-\zeta(\theta,u)\sin[2\theta]}e^{-i\zeta(\theta,u)(-\eta(u)+\cos[2\theta])}e^{-il(2\theta+\frac{3\pi}{2})}\cos[Q_{-}u-\frac{1}{2}\kappa(l,Q_{-})]];$
$\displaystyle
U_{\epsilon(l,Q_{-})\downarrow}(\theta,u)=G^{\frac{-1}{4}}(\theta,u)\cdot[U^{(even,k)}_{\epsilon(l,Q_{-})\downarrow}(\theta,u)+U^{(odd,k)}_{\epsilon(l,Q_{-})\uparrow}(\theta,u)];$
$\displaystyle
U^{(even,k)})_{\epsilon(l,Q_{-})\downarrow}(\theta,u)=2ie^{\frac{-i}{2}\kappa(l,Q_{-})}[e^{\zeta(\theta,u)\sin[2\theta]}e^{i\zeta(\theta,u)(\eta(u)-\cos[2\theta])}e^{i((l+1)(2\theta+\frac{3\pi}{2}))}\sin[Q_{-}u-\frac{1}{2}\kappa(l,Q_{-})]$
$\displaystyle-(-1)^{l}e^{-\zeta(\theta,u)\sin[2\theta]}e^{i\zeta(\theta,u)(-\eta(u)+\cos[2\theta])}e^{-i((l+1)(2\theta+\frac{3\pi}{2}))}\sin[Q_{-}u+\frac{1}{2}\kappa(l,Q_{-})]];$
$\displaystyle
U^{(odd,k)}_{\epsilon(l,Q_{-})\downarrow}(\theta,u)=2e^{\frac{-i}{2}\kappa(l,Q_{-})}[e^{\zeta(\theta,u)\sin[2\theta]}e^{i\zeta(\theta,u)(\eta(u)-\cos[2\theta])}e^{i((l+1)(2\theta+\frac{3\pi}{2}))}\cos[Q_{-}u-\frac{1}{2}\kappa(l,Q_{-})]$
$\displaystyle-(-1)^{l}e^{-\zeta(\theta,u)\sin[2\theta]}e^{i\zeta(\theta,u)(-\eta(u)+\cos[2\theta])}e^{-i((l+1)(2\theta+\frac{3\pi}{2}))}\cos[Q_{-}u+\frac{1}{2}\kappa(l,Q_{-})]];$

Similarly for $Q_{+}$we obtain the wave function:

$\displaystyle
U_{\epsilon(l,Q_{+})\uparrow}(\theta,u)=G^{\frac{-1}{4}}(\theta,u)\cdot[U^{(even,k)}_{\epsilon(l,Q_{+})\uparrow}(\theta,u)+U^{(odd,k)}_{\epsilon(l,Q_{+})\uparrow}(\theta,u)];$
$\displaystyle
U^{(even,k)}_{\epsilon(l,Q_{+})\uparrow}(\theta,u)=2ie^{\frac{-i}{2}\kappa(l,Q_{+})}[(-1)^{l}e^{-\zeta(\theta,u)\sin[2\theta]}e^{-i\zeta(\theta,u)(-\eta(u)+\cos[2\theta])}e^{il(2\theta+\frac{3\pi}{2})}\sin[Q_{+}u+\frac{1}{2}\kappa(l,Q_{+})]$
$\displaystyle+e^{\zeta(\theta,u)\sin[2\theta]}e^{-i\zeta(\theta,u)(\eta(u)-\cos[2\theta])}e^{-il(2\theta+\frac{3\pi}{2})}\sin[Q_{+}u-\frac{1}{2}\kappa(l,Q_{+})]];$
$\displaystyle
U^{(odd,k)}_{\epsilon(l,Q_{+})\uparrow}(\theta,u)=2e^{\frac{-i}{2}\kappa(l,Q_{+})}[(-1)^{l}e^{-\zeta(\theta,u)\sin[2\theta]}e^{-i\zeta(\theta,u)(-\eta(u)+\cos[2\theta])}e^{il(2\theta+\frac{3\pi}{2})}\cos[Q_{+}u+\frac{1}{2}\kappa(l,Q_{+})]$
$\displaystyle+e^{\zeta(\theta,u)\sin[2\theta]}e^{-i\zeta(\theta,u)(\eta(u)-\cos[2\theta])}e^{-il(2\theta+\frac{3\pi}{2})}\cos[Q_{+}u-\frac{1}{2}\kappa(l,Q_{+})]];$
$\displaystyle
U_{\epsilon(l,Q_{+})\downarrow}(\theta,u)=G^{\frac{-1}{4}}(\theta,u)\cdot[U^{(even,k)}_{\epsilon(l,Q_{+})\downarrow}(\theta,u)+U^{(odd,k)}_{\epsilon(l,Q_{+})\downarrow}(\theta,u)];$
$\displaystyle
U^{(even,k)}_{\epsilon(l,Q_{+})\downarrow}(\theta,u)=2ie^{\frac{-i}{2}\kappa(l,Q_{+})}[-(-1)^{l}e^{-\zeta(\theta,u)\sin[2\theta]}e^{i\zeta(\theta,u)(-\eta(u)+\cos[2\theta])}e^{i(l+1)(2\theta+\frac{3\pi}{2})}\sin[Q_{+}u-\frac{1}{2}\kappa(l,Q_{+})]$
$\displaystyle+e^{\zeta(\theta,u)\sin[2\theta]}e^{i\zeta(\theta,u)(\eta(u)-\cos[2\theta])}e^{-i(l+1)(2\theta+\frac{3\pi}{2})}\sin[Q_{+}u+\frac{1}{2}\kappa(l,Q_{+})]];$
$\displaystyle
U^{(odd,k)}_{\epsilon(l,Q_{+})\downarrow}(\theta,u)=2e^{\frac{-i}{2}\kappa(l,Q_{+})}[-(-1)^{l}e^{-\zeta(\theta,u)\sin[2\theta]}e^{i\zeta(\theta,u)(-\eta(u)+\cos[2\theta])}e^{i(l+1)(2\theta+\frac{3\pi}{2})}\cos[Q_{+}u-\frac{1}{2}\kappa(l,Q_{+})]$
$\displaystyle+e^{\zeta(\theta,u)\sin[2\theta]}e^{i\zeta(\theta,u)(\eta(u)-\cos[2\theta])}e^{-i(l+1)(2\theta+\frac{3\pi}{2})}\cos[Q_{+}u+\frac{1}{2}\kappa(l,Q_{+})]];$

where $G^{\frac{-1}{4}}(\theta,u)$ is the Jacobian transformation induced by
the metric tensor.

IV -Computation of the STM density of states

A-Description of the STM method

The STM tunneling current $I$ is a function of the bias voltage $V$ which
gives spatial and spectroscopic information about the electronic surface
states. At zero temperature, the derivative of the current with respect the
bias voltage $V$ is given in term of the single particles eigenvalues:
$\epsilon(m,q_{-})=\pm\hbar v_{F}\sqrt{(\frac{2\pi}{L}m)^{2}+q_{-}^{2}}$,
$\epsilon(m,q_{+})=\pm\hbar v_{F}\sqrt{(\frac{2\pi}{L}m)^{2}+q_{-}^{2}}$
,$m=0,1,2,3...$ for contour $n=0$. For the upper and lower circular contours
$n=\pm 1$, we have :$\epsilon(l,Q_{-})=\pm\hbar
v_{F}\sqrt{(\frac{l+\frac{1}{2}}{R_{g}(1)})^{2}+Q_{-}^{2}}$
,$\epsilon(l,Q_{+})=\pm\hbar
v_{F}\sqrt{(\frac{l+\frac{1}{2}}{R_{g}(1)})^{2}+Q_{+}^{2}}$ ,$l=0,1,2,3..$.
The $STM$ density of states is computed for a voltage $V$ between the $STM$
tip and the sample. The tunneling current is a function of the bias voltage
$V$ and the chemical potential $\mu>0$ kittel :

$\displaystyle\frac{dI}{dV}\propto
D(E=eV;s,u)\equiv\sum_{n}D^{(n)}(E=eV;s,u)=$
$\displaystyle=\sum_{\eta=\pm}[\sum_{m}\sum_{q_{r}=q_{+},q_{-}}\sum_{\sigma}|U^{(n=0;m,q_{r})}_{\sigma}(x,y)|^{2}\delta[eV+\mu-\eta\hbar
v_{F}\sqrt{(\frac{2\pi}{L}m)^{2}+q_{r}^{2}}]$ $\displaystyle+\sum_{n=\pm
1}\sum_{l}\sum_{Q_{r}=Q_{+},Q_{-}}\sum_{\sigma}|U^{(n=\pm
1;l,Q_{r})}_{\sigma}(\theta,u)|^{2}\delta[eV+\mu-\eta\hbar
v_{F}\sqrt{(\frac{l+\frac{1}{2}}{R_{g}(1)})^{2}+Q_{r}^{2}}]]$

($\eta=+$ corresponds to electrons with energy $0<\epsilon\leq\mu$ and
$\eta=-$ corresponds to electrons below the Dirac point $\epsilon<0$. For the
rest of this paper we will take the chemical potentials to be $\mu=120mV$
(this is typical value for the $TI$ ). We will neglect the states with
$\eta=-$ which correspond to particles below the Dirac cone. The density of
states at the tunneling energy $eV$ is weighted by the probability density of
the $STM$ tip at position $[x,y]$ for n=0. The contours for $n=\pm 1$ will be
parametrized in terms of the polar angle $\theta$ and transverse coordinate
$u$. The proportionality factor $J$ for the tunneling probability (not shown
in the equation ) $\frac{dI}{dV}=JD(V;x,y)$ is a function of the distance
between the tip and the sample. The notation $D^{(n)}(V;x,y)$ represents the
tunneling density for the different contours.

IVB-The tunneling density of states $D^{(n=0)}(V;x,y)$ for $n=0$

Summing up the single particle states weighted with occupation probability
$|U^{(n=0;m,q_{r})}_{\sigma}(x,y)|^{2}$, we obtain a space dependent density
of states for the two dimensional boundary surface ,$\frac{-L}{2}\leq
x\leq\frac{L}{2}$ and the coordinate $y$ is restricted to the regions
$\frac{d}{2}<y\leq\frac{L}{2}$ and $\frac{-L}{2}<y\leq\frac{d}{2}$. We will
perform the computation at the thermodynamic limit, namely we replace the
discrete momentum $\frac{\pi}{L}k$ by $Y=\frac{k}{N}$ and $\frac{2\pi}{L}m$ by
$X=\frac{m}{N}$ where $N=\frac{L}{a}$. We find for the dimensionless momentum
$\hat{q}\equiv qa$ the equations : $\hat{q}_{\pm}(Y)=\pi
Y\pm\frac{1}{N}\tan^{-1}[\frac{\hat{q}_{\pm}(Y)}{2\pi X}]$ where $2\pi
X=pa=\hat{p}$. As a result we obtain the following density of states
$\frac{\partial\hat{q}_{\pm}}{\partial Y}$

$\displaystyle[\frac{\partial\hat{q}_{+}}{\partial
Y}]^{-1}=\frac{1}{\pi}\frac{\hat{q}_{+}^{2}+\hat{p}^{2}-\frac{1}{N}\hat{p}}{\hat{q}_{+}^{2}+\hat{p}^{2}}$
$\displaystyle[\frac{\partial\hat{q}_{-}}{\partial
Y}]^{-1}=\frac{1}{\pi}\frac{\hat{q}_{-}^{2}+\hat{p}^{2}+\frac{1}{N}\hat{p}}{\hat{q}_{-}^{2}+\hat{p}^{2}}$

Using this results, we compute the tunneling density of states in terms of the
energy $\mu+eV$ measured with respect the chemical potential $\mu$ and the
transverse energy $\epsilon_{\bot}\equiv\hbar v_{F}q_{\pm}$.

$\displaystyle
D^{(n=0)}(V;x,y)=(\frac{L}{hv_{F}})^{2}(\frac{B^{(2)}}{L})^{2}\frac{1}{4\sqrt{G(x,y)}}e^{\frac{-B^{(2)}}{\pi}(\frac{x}{x^{2}+y^{2}+a^{2}})}[\int_{0}^{E_{max.}}\,d\epsilon_{\bot}\frac{(\mu+eV)}{\sqrt{(\mu+eV)^{2}-\epsilon_{\bot}^{2}}}\dot{}$
$\displaystyle[\frac{1}{2}(1+\frac{1}{\pi}\frac{hv_{F}}{L(\mu+V)}\sqrt{1-(\frac{\epsilon_{\bot}}{\mu+V})^{2}}\hskip
7.22743pt)+\frac{1}{2}(1-\frac{1}{\pi}\frac{hv_{F}}{L(\mu+V)}\sqrt{1-(\frac{\epsilon_{\bot}}{\mu+V})^{2}}\hskip
7.22743pt)]$
$\displaystyle+\frac{hv_{F}}{L}(H[\mu+V-\frac{hv_{F}}{2L}]-H[\mu+eV-
E_{max}])\cdot((\cos[\frac{(\mu+eV)}{\hbar
v_{F}}y-\frac{\pi}{4}])^{2}+(\cos[\frac{(\mu+eV)}{\hbar
v_{F}}y-\frac{\pi}{4}])^{2})]$
$\displaystyle=(\frac{L}{hv_{F}})^{2}(\frac{B^{(2)}}{L})^{2}\frac{1}{4\sqrt{G(x,y)}}e^{\frac{-B^{(2)}}{\pi}(\frac{x}{x^{2}+y^{2}+a^{2}})}[\int_{0}^{E_{max.}}\,d\epsilon_{\bot}\frac{(\mu+eV)}{\sqrt{(\mu+eV)^{2}-\epsilon_{\bot}^{2}}}$
$\displaystyle+\frac{hv_{F}}{L}(H[\mu+V-\frac{hv_{F}}{2L}]-H[\mu+eV-
E_{max}])\cdot((\cos[\frac{(\mu+eV)}{\hbar
v_{F}}y-\frac{\pi}{4}])^{2}+(\cos[\frac{(\mu+eV)}{\hbar
v_{F}}y-\frac{\pi}{4}])^{2})]=$
$\displaystyle(\frac{L}{hv_{F}})^{2}(\frac{B^{(2)}}{L})^{2}\frac{1}{4\sqrt{G(x,y)}}e^{\frac{-B^{(2)}}{\pi}(\frac{x}{x^{2}+y^{2}+a^{2}})}[\frac{\pi}{2}(\mu+eV)+\frac{hv_{F}}{L}(H[\mu+V-\frac{hv_{F}}{2L}]-H[\mu+eV-
E_{max}])]$ $\displaystyle for\hskip 14.45377pt|y|>d$

$H[\mu+eV-\frac{hv_{F}}{2L}]$ is the step function which is one for
$\mu+eV-\frac{hv_{F}}{2L}\geq 0$ and zero otherwise. $a=\frac{2\pi}{\Lambda}$
is the short distance cut-off and $E_{max}=\hbar v_{F}\Lambda<0.3eV$ is the
maximal energy which restricts the validity of the Dirac model. We observe in
the second line that the asymmetry in the density of states
$1\pm\frac{1}{\pi}\frac{hv_{F}}{L(\mu+V)}\sqrt{1-(\frac{\epsilon_{\bot}}{\mu+V})^{2}}\hskip
7.22743pt)$ cancels.

Equation $(51)$ shows that the tunneling density of states is linear in the
energy $\mu+eV$ (in the present case we have looked only for energies above
the Dirac cone ). For the chemical potential $\mu=120mV$, the zero energy
corresponds to the Voltage $V=-120mV$. The tunneling density of states has a
constant part at energies $\frac{hv_{F}}{2L}\approx 0.2mV$ for
$-120mV<V<-119.8mV$. For $V>-119.8mV$ the density of states is proportional to
$\mu+eV$.

In figure $2$ we have plotted the tunneling density of states as a function of
the coordinates $x$ and $y$. The shape of the plot is governed by the the
multiplicative factor $e^{-\frac{B^{(2)}}{\pi}(\frac{x}{x\pm iy})}$ which
governs the solutions in eq.$(35)$. We observe that the density of state is
maximal in the region $|y|<10B^{(2)}$.

Figure $3$ shows the dependence on the voltage $V$ and coordinate $y$. We
observe the linear increase in the tunneling density of states which is
maximal in the region $|y|<10B^{(2)}$.

IVC-The tunneling density of states
$D^{(n=0)}(V,x,y;\vec{r}_{1},..\vec{r}_{2M})$ for $2M$ dislocations.

For many dislocations which satisfy $\sum_{w=1}^{2M}B^{(2,w)}=0$ ( sum of the
Burger vectors is zero ) with the core centered at $[x_{w},y_{w}]$
,$w=1,2..2M$ the coordinate $\vec{r}=(x,y)\rightarrow[X(\vec{r}),Y(\vec{r})]$
is replaced by
$[X(\vec{r})=x,Y(\vec{r})=y+\sum_{w}\frac{B^{(2,w)}}{2\pi}\tan^{-1}(\frac{y-y_{w}}{x-x_{w}})]$.
Following the method used previously, we find the edge Hamiltonian with many
dislocations takes the form:

$h^{edge}(w=1,2...2M)\approx
i\sigma^{1}[\partial_{y}-\frac{i}{2}\sum_{w=1}^{2M}\sigma^{3}B^{(2,w)}\delta^{2}(\vec{r}-\vec{r}_{w})]-i\sigma^{2}\partial_{x}$
(52)

As a result, the wave functions are given by:

$\displaystyle
U^{(n=0,w=1,2...2M)}_{\uparrow}(x,y)\propto\prod_{w=1,2...2M}e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{(x-x_{w})+i(y-y_{w})})}F^{(n=0)}_{\uparrow}(x,y)$
$\displaystyle
U^{(n=0,w=1,2...2M)}_{\downarrow}(x,y)\propto\prod_{w=1,2...2M}e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{(x-x_{w})-i(y-y_{w})})}F^{(n=0)}_{\downarrow}(x,y)$
(53)

Using these wave functions, we find that the tunneling density of states is
given by:

$D^{(n=0)}(V,x,y;\vec{r}_{1},..\vec{r}_{2M})\propto\prod_{w=1,2...2M}e^{-\frac{B^{(2)}}{\pi}(\frac{(x-x_{w})}{(x-x_{w})^{2}+(y-y_{w})^{2}+a^{2}})}$
(54)

In figure $4$ we show the tunneling density of states for an even number of
dislocations in the $y$ directions which have the core on the $y=0$ axes
($\vec{r}_{w}=[x_{w},y_{w}=0]$, $w=1,2,3,...2M$). When the coordinate of the
$w=1,2,3,...2M$ dislocations is replaced by a continuum variable $w$ which can
be described by a domain a wall model: $h^{domain-wall}(x,y)=\hbar
v_{F}[-i\sigma^{1}\partial_{y}+i\sigma^{2}\partial_{x}-\sigma^{3}\kappa M(y)]$
where $M(y)=sgn[y]|M(y)|$ Jackiw .

Using this model find that the tunneling density of states density $D^{domain-
wall}(V;x,y)$ confined to $|y|<W$ (the width $W$ depends on the explicit form
of the domain wall function $M(y)$ and strength $\kappa$) is given by:
$D^{domain-
wall}(V;x,y)\propto(\frac{L}{hv_{F}})^{2}e^{-2\kappa\int_{0}^{|y|}\,dy^{\prime}M(y^{\prime})}$.
This show the similarity between the result obtain from the $domain-wall$
model and the large numbers of of dislocations given in equation $(54)$.

IVD-The tunneling density of states $D^{(n=\pm 1)}(V;\theta,u)$ for the $n=\pm
1$ contours.

Following the same procedure as used for the $n=0$ and using the
eigenfunctions for $n=\pm 1$ we find :

$D^{(n=\pm 1)}(V;\theta,u)\equiv D^{(n=\pm
1)}(\mu,V;\theta,u)_{even}+D^{(n=\pm 1)}(\mu,V;\theta,u,\mu)_{odd}$ (55)

For the even $k$’s, we solve for the momentum $Q_{+}$ and $Q_{-}$ and find:

$\displaystyle D^{(n=\pm 1)}(\mu,V;\theta,u)_{even}=\frac{(B^{(2)})^{2}}{2\pi
R_{g}(1)D(1)\sqrt{G(\theta,u)}}\sum_{Q_{r}=Q_{+},Q_{-}}\sum_{l=0}^{\infty}\delta[eV+\mu-\hbar
v_{F}\sqrt{(\frac{l+\frac{1}{2}}{R_{g}(1)})^{2}+Q_{r}^{2}}]$
$\displaystyle[(e^{-2\zeta(\theta,u)\sin[2\theta]}+e^{2\zeta(\theta,u)\sin[2\theta]})((\sin[Q_{r}u-\frac{1}{2}\kappa(l,Q_{r})])^{2}+(\sin[Q_{r}u+\frac{1}{2}\kappa(l,Q_{r})])^{2})+$
$\displaystyle
2(-1)^{l}\sin[Q_{r}u+\frac{1}{2}\kappa(l,Q_{r})]\sin[Q_{r}u-\frac{1}{2}\kappa(l,Q_{r})]\cdot$
$\displaystyle(\cos[l(\theta+\frac{3\pi}{2})-\zeta(\theta,u)(-\eta(u)+\cos[2\theta])]-\cos[(l+1)(\theta+\frac{3\pi}{2})+\zeta(\theta,u)(-\eta(u)+\cos[2\theta])]];$

Similarly for the odd $k$’s we find:

$\displaystyle D^{(n=\pm 1)}(\mu,V;\theta,u)_{odd}=\frac{(B^{(2)})^{2}}{2\pi
R_{g}(1)D(1)\sqrt{G(\theta,u)}}\sum_{Q_{r}=Q_{+},Q_{-}}\sum_{l=0}^{\infty}\delta[eV+\mu-\hbar
v_{F}\sqrt{(\frac{l+\frac{1}{2}}{R_{g}(1)})^{2}+Q_{r}^{2}}]$
$\displaystyle[(e^{-2\zeta(\theta,u)\sin[2\theta]}+e^{2\zeta(\theta,u)\sin[2\theta]})((\cos[Q_{r}u-\frac{1}{2}\kappa(l,Q_{r})])^{2}+(\cos[Q_{r}u+\frac{1}{2}\kappa(l,Q_{r})])^{2})+$
$\displaystyle
2(-1)^{l}\cos[Q_{r}u+\frac{1}{2}\kappa(l,Q_{r})]\cos[Q_{r}u-\frac{1}{2}\kappa(l,Q_{r})]\cdot$
$\displaystyle(\cos[l(\theta+\frac{3\pi}{2})-\zeta(\theta,u)(-\eta(u)+\cos[2\theta])]-\cos[(l+1)(\theta+\frac{3\pi}{2})+\zeta(\theta,u)(-\eta(u)+\cos[2\theta])]]$

For the present case the energy scale of the excitations is governed by the
radius $R_{g}(1)$ and width $D$. The spectrum is discrete and we can’t replace
it by a continuum density of states as we did for the case $n=0$.

In figure $5$ we show the tunneling density of states at a fixed polar angle
$\theta=\frac{\pi}{2}$ as a function of the voltage $V$. We observe that the
density of states is dominated by high energy eigenvalues. This solutions are
localized in energy. The range of the spectrum is above $\mu+eV>200mV$ which
is well separated from the low energy spectrum controlled by the $n=0$ contour
(which ranges from $-120mV$ to $70mV$).

Figure $6$ shows the tunneling density of states as a function of the polar
angle $\theta$ for a fixed energy . The periodicity in $\theta$ is controlled
by the discrete energy eigenvalues.

In figure $7$ we show the tunneling density of states at a fixed voltage $V$
as a function of the polar angle $0<\theta<\pi$ and width $|u|<0.1$.

V-The charge current-the in plane spin on the surface of the $h^{T.I.}$
Hamiltonian

A-The current in the absence of the edge dislocation for the $h^{T.I.}$

From the Hamiltonian given in equation $1$ we compute the equation of motion
for the velocity operator:
$\frac{dx}{dt}=\frac{1}{i\hbar}[x,h]=v_{F}\sigma^{y}$ ,
$\frac{dy}{dt}=\frac{1}{i\hbar}[y,h]=-v_{F}\sigma^{x}$. We multiply the
velocity operator by the charge $(-e)$ and identify the charge current
operators : $\hat{J}_{x}=(-e)v_{F}\sigma^{2}$,
$\hat{J}_{y}=(-e)(-v_{F})\sigma^{1}$. This also represent the ”‘real”’ spin on
the surface. Therefore, the charge current is a measure of the in-plane spin
on the surface.

Integrating over the $y$ coordinate we obtain the current $I_{x}^{T.I.}$ in
the $x$ direction. Using the eigenstates $\Gamma_{p,q}(x,y)$ and
$\Gamma_{-p,q}(x,y)$ of the $h^{T.I.}$ Hamiltonian

$\Gamma_{p,q}(x,y)=e^{ipx}e^{iqy}\left(\begin{array}[]{cc}1\\\
ie^{i\chi(p,q)}\end{array}\right)$
$\Gamma_{-p,q}(x,y)=e^{-ipx}e^{iqy}\left(\begin{array}[]{cc}1\\\
-ie^{-i\chi(p,q)}\end{array}\right)$

we find
$(\Gamma_{p,q}(x,y))(\sigma^{2})(\Gamma_{p,q}(x,y))=-(\Gamma_{-p,q}(x,y))(\sigma^{2})(\Gamma_{-p,q}(x,y))$
therefore, we conclude that the current $I_{x}^{T.I.}=0$ is zero.

VB-The current in the presence of the edge dislocation

We will compute the current in the presence of the edge dislocation. The
current operator $\hat{J}^{edge}_{x}(x,y)$ will be given in terms of the
transformed currents. We find that the current density operator
$J^{edge}_{x}(x,y)$ is given by:

$\hat{J}^{edge}_{x}(x,y)=(-e)v_{F}[\sigma^{2}e^{x}_{1}-\sigma^{1}e^{x}_{2}]=(-e)v_{F}\sigma^{2}-(-e)v_{F}\frac{B^{(2)}}{2\pi}(\frac{y\sigma^{1}+x\sigma^{2}}{x^{2}+y^{2}})\approx(-e)v_{F}\sigma^{2}$
(58)

We use the zero order current operator
$\hat{J}^{edge}_{x}(x,y)\approx(-e)v_{F}\sigma^{2}$ to construct the second
quantization form for the current density. The operator is defined with
respect the to shifted ground state $|\mu>\equiv|\tilde{0}>$ with the energy
$E=\epsilon-\mu$ measured with respect the chemical potential and spinor field
$\Psi_{n=0}(x,y)$.

$J^{edge}_{x}(x,y)=<\mu|\Psi_{n=0}^{\dagger}(x,y)\hat{J}^{edge}_{x}(x,y)\Psi_{n=0}(x,y)|\mu>$
(59)

Using the spinor eigenfunction given in equation $(35)$ and the second
quantized form with the electron like operators $\alpha_{E,R}$,$\alpha_{E,L}$
and hole like $\beta_{E,R}$,$\beta_{E,L}$ we find :

$\Psi_{n=0}(x,y;t)\approx\sum_{E>0}[\alpha_{E,R}\left(\begin{array}[]{cc}U^{(n=0,R)}_{\uparrow}(x,y)\\\
U^{(n=0,R)}_{\downarrow}(x,y)\end{array}\right)_{E+\mu}{e^{-i\frac{E}{\hbar}t}}+\beta^{\dagger}_{E,R}\left(\begin{array}[]{cc}U^{(n=0,R)}_{\uparrow}(x,y)\\\
U^{(n=0,R)}_{\downarrow}(x,y)\end{array}\right)_{-E+\mu}e^{i\frac{E}{\hbar}t}$
$+\alpha_{E,L}\left(\begin{array}[]{cc}U^{(n=0,L)}_{\uparrow}(x,y)\\\
U^{(n=0,L)}_{\downarrow}(x,y)\end{array}\right)_{E+\mu}e^{-i\frac{E}{\hbar}t}+\beta^{\dagger}_{E,L}\left(\begin{array}[]{cc}U^{(n=0,L)}_{\uparrow}(x,y)\\\
U^{(n=0,L)}_{\downarrow}(x,y)\end{array}\right)_{-E+\mu}e^{i\frac{E}{\hbar}t}]$
(60)

The current is a sum of two terms computed with the eigen spinor obtained in
equation $(35)$:
$[U^{(n=0,R)}_{\uparrow}(x,y),U^{(n=0,R)}_{\downarrow}(x,y)]^{T}\sigma^{2}[U^{(n=0,R)}_{\uparrow}(x,y),U^{(n=0,R)}_{\downarrow}(x,y)]$
and
$[U^{(n=0,L)}_{\uparrow}(x,y),U^{(n=0,L)}_{\downarrow}(x,y)]^{T}\sigma^{2}[U^{(n=0,L)}_{\uparrow}(x,y),U^{(n=0,L)}_{\downarrow}(x,y)]$
which have opposite signs. Due to the parity violation caused by the
dislocation, the density of states is asymmetric
$1\pm\frac{1}{\pi}\frac{hv_{F}}{L(\mu+V)}\sqrt{1-(\frac{\epsilon_{\bot}}{\mu+V})^{2}}\hskip
7.22743pt)$ resulting in a finite current. We integrate over the transversal
direction $y$ and obtain the edge current $I_{x}^{n=0,edge}$.

$\displaystyle
I_{x}^{n=0,edge}=(-e)v_{F}\int_{-\frac{L}{2}}^{\frac{L}{2}}\frac{dx}{L}\int_{-\frac{L}{2}}^{\frac{L}{2}}\,dy<\mu|J^{edge}_{x}(x,y)|\mu>=$
$\displaystyle\frac{(-e)v_{F}}{4\pi}(\frac{L}{hv_{F}})^{2}(\frac{1}{L})\int_{-\frac{L}{2}}^{\frac{L}{2}}\frac{dx}{L}\int_{-\frac{L}{2}}^{\frac{L}{2}}\frac{dy}{L}\frac{e^{\frac{-B^{(2)}}{\pi}(\frac{x}{x^{2}+y^{2}+a^{2}})}}{\sqrt{G(x,y)}}\int\,d\epsilon_{||}\int
d\epsilon_{\bot}H[\mu-\sqrt{(\epsilon_{||})^{2}+(\epsilon_{\bot})^{2}}\hskip
7.22743pt]\frac{(hv_{F}/L)\cdot\epsilon_{||}}{(\epsilon_{||})^{2}+(\epsilon_{\bot})^{2}}$
$\displaystyle=\frac{1}{4\pi}(\frac{-ev_{F}}{L})(\frac{\mu}{hv_{F}/L})f[\frac{B^{(2)}}{L}]\cdot(H[\mu+eV-\frac{hv_{F}}{L}]-H[\mu+eV-
E_{max.}]);\hskip 14.45377ptf[\frac{B^{(2)}}{L}]\approx 6.22$

$H[\mu-\sqrt{(\epsilon_{||})^{2}+(\epsilon_{\bot})^{2}}\hskip 7.22743pt]$ is
the step function which is one for
$\sqrt{(\epsilon_{||})^{2}+(\epsilon_{\bot})^{2}}\leq\mu$. The single particle
energies are $\epsilon_{\bot}=\hbar v_{F}q_{\pm}$ and $\epsilon_{||}=\hbar
v_{F}p$. For $L\approx 10^{-6}m$, chemical potential $\mu=120mV$ and
$\frac{L}{B^{(2)}}\approx 100$ we find that the current $I_{x}^{n=0,edge}$ is
in the range of $mA$.

To conclude, we have shown that the presence of an edge dislocation gives rise
to a non-zero current which is a manifestation of the in-plane component of
the spin on the two dimensional surface . Therefore a nonzero value
$I_{x}^{n=0,edge}\neq 0$ will be an indication of the presence of the edge
dislocation. This effect might be measured using a coated tip with magnetic
material used by the technique of Magnetic Force Microscopy.

VI-Conclusions

We have used the coordinate transformation method to investigate $TI$ in the
presence of deformations. We have computed the spin connection and the metric
tensor for the three dimensional $TI$. This theory has been applied to the
surface of a $TI$ with an edge dislocation. We have shown that the tunneling
density of states is confined to two dimensional region $n=0$ and to high
energy circular contours with $n=\pm 1$. The edge dislocations violate the
parity symmetry. As a result a current which is a manifestation of in plane
spin orientation is generated. The in plane spin orientation is a
manifestation of the parity violation induced by the edge dislocation. We
propose that scanning tunneling methods might be able to verify our
prediction.

Appendix -A

We consider that a two dimensional manifold with a mapping from the curved
space $X^{a}$, $a=1,2$, to the $local$ $flat$ space $x^{\mu}$, $\mu=x,y$
exists. We introduce the tangent vector Green
$e^{a}_{\mu}(\vec{x})=\frac{\partial X^{a}(\vec{x})}{\partial x^{\mu}}$,
$\mu=x,y$ which satisfies the orthonormality relation
$e^{a}_{\mu}(\vec{x})e^{b}_{\mu}(\vec{x})=\delta_{a,b}$ (here we use the
convention that we sum over indices which appear twice). The metric tensor for
the curved space is given in terms of the flat metric $\delta_{a,b}$ and the
scalar product of the tangent vectors:
$e^{a}_{\mu}(\vec{x})e^{a}_{\nu}(\vec{x})=g_{\mu,\nu}(\vec{x})$. The linear
connection is determined by the Christoffel tensor
$\Gamma^{\lambda}_{\mu,\nu}$ :

$\nabla_{\partial_{\mu}}\partial_{\nu}=-\Gamma^{\lambda}_{\mu,\nu}\partial_{\lambda}$
(62)

The Christoffel tensor is constructed from the metric tensor
$g_{\mu,\nu}(\vec{x})$.

$\Gamma^{\lambda}_{\mu,\nu}=-\frac{1}{2}\sum_{\tau=x,y}g^{\lambda,\tau}(\vec{x})[\partial_{\nu}g_{\nu,\tau}(\vec{x})+\partial_{\mu}g_{\nu,\tau}(\vec{x})-\partial_{\tau}g_{\mu,\nu}(\vec{x})]$
(63)

Next, we introduce the vector field
$\vec{V}=V^{a}\partial_{a}=V^{\mu}\partial_{\mu}$ where $a=1,2$ are the
components in the curved space and $\mu=x,y$ represents the coordinate in the
fixed cartesian frame. The covariant derivative of the vector field $V^{a}$ is
determined by the spin connection $\omega_{q,b}^{\mu}$ which needs to be
computed:

$D_{\mu}V^{a}(\vec{x})=\partial_{\mu}V^{a}(\vec{x})+\omega_{a,b}^{\mu}V^{b}$
(64)

For a two component spinor, we can identify the spin connection in the
following way: The spinor in the the curved space (generated by the
dislocation) is represented by $\widetilde{\Psi}(\vec{X})$ and in the
Cartesian space it is given by is given by $\Psi(\vec{x})$ Maggiore . The two
component spinor represents a chiral fermion which transform under spatial
rotation as spin half fermion:

$\displaystyle\widetilde{\Psi}(\vec{X})=e^{\frac{-i}{2}\omega_{1,2}\sigma_{3}}\Psi(\vec{x})$
$\displaystyle e^{\frac{-i}{2}\omega_{1,2}\sigma_{3}}\equiv
e^{\frac{1}{2}\omega_{a,b}\Sigma^{a,b}}\equiv
e^{\sum_{a=1,2}\sum_{b=1,2}\frac{1}{2}\omega_{a,b}\Sigma^{a,b}}$
$\displaystyle\omega_{a,b}\equiv-\omega_{b,a}$
$\displaystyle\Sigma^{a,b}\equiv\frac{1}{4}[\sigma^{a},\sigma^{b}]$

We have used the anti symmetric property of the rotation matrix
$\omega_{a,b}\equiv-\omega_{b,a}$, and the representation of the generator
$\Sigma^{a,b}$ in terms of the Pauli matrices.

Therefore for a two component spinor we obtain the connection:

$D_{\mu}\Psi(\vec{x})=(\partial_{\mu}+\frac{1}{2}\omega^{a,b}_{\mu}\Sigma_{a,b})\Psi(\vec{x})\equiv(\partial_{\mu}+\frac{1}{8}\omega^{a,b}_{\mu}[\sigma_{a},\sigma_{b}])\Psi(\vec{x})$
(66)

Next we will compute the spin connection $\omega^{a,b}_{\mu}$ using the
Christoffel tensor. In the physical coordinate basis $x^{\mu}$ the covariant
derivative $D_{\mu}V^{\nu}(\vec{x})$ is determined by the Christoffel tensor:

$D_{\mu}V^{\nu}(\vec{x})=\partial_{\mu}V^{\nu}(\vec{x})+\Gamma^{\lambda}_{\mu,\nu}V^{\lambda}$
(67)

The relation between the spin connection and the linear connection can be
obtained from the fact that the two covariant derivative of the vector
$\vec{V}$ are equivalent.

$D_{\mu}V^{a}=e^{a}_{\nu}D_{\mu}V^{\nu}$ (68)

Since we have the relation $V^{a}=e^{a}_{\nu}V^{\nu}$ it follows from the last
equation

$D_{\mu}[e^{a}_{\nu}]=D_{\mu}\partial_{\nu}e^{a}=(D_{\mu}\partial_{\nu})e^{a}+\partial_{\nu}(D_{\mu}e^{a})=0$
(69)

Using the definition of the Christoffel index and the differential geometry
relation
$\nabla_{\partial_{\mu}}\partial_{\nu}=-\Gamma^{\lambda}_{\mu,\nu}\partial_{\lambda}$
Green , we obtain the relation between the spin connection and the linear
connection:

$D_{\mu}[e^{a}_{\nu}]=\partial_{\mu}e^{a}_{\nu}(\vec{x})-\Gamma^{\lambda}_{\mu,\nu}e^{a}_{\lambda}(\vec{x})+\omega^{a}_{\mu,b}e^{b}_{\nu}(\vec{x})\equiv
0$ (70)

Solving this equation, we obtain the spin connection given in terms of the
Burger vector. We multiply from left equation $(70)$ by the tangent vector
$e^{a}_{\nu}$ and replace $\Gamma^{\lambda}_{\mu,\nu}$ with the representation
given in equation $(63)$. We use the metric tensor relations
$e^{a}_{\mu}(\vec{x})e^{b}_{\mu}(\vec{x})=\delta_{a,b}$,
$e^{a}_{\mu}(\vec{x})e^{a}_{\nu}(\vec{x})=g_{\mu,\nu}(\vec{x})$. and find
Green :

$\displaystyle\omega^{a,b}_{\mu}=\frac{1}{2}e^{\nu,a}(\partial_{\mu}e^{b}_{\nu}-\partial_{\nu}e^{b}_{\mu})-\frac{1}{2}e^{\nu,b}(\partial_{\mu}e^{a}_{\nu}-\partial_{\nu}e^{a}_{\mu})$
$\displaystyle-\frac{1}{2}e^{\rho,a}e^{\sigma,b}(\partial_{\rho}e_{\sigma,c}-\partial_{\sigma}e_{\rho,c})e^{c}_{\mu}$
(71)

We notice the asymmetry between $e^{\nu,a}$ and $e_{a,\nu}$:

$e^{\nu,a}\equiv g^{\nu,\lambda}e^{a}_{\lambda}$ and
$e_{a,\nu}\equiv\delta_{a,b}e^{b}_{\nu}$

For our case we have a two component the spin connection $\omega_{x}^{12}$ and
$\omega_{y}^{12}$

$\displaystyle\omega_{x}^{12}=\frac{1}{2}e^{\nu,1}(\partial_{x}e^{2}_{\nu}-\partial_{\nu}e^{2}_{x})-\frac{1}{2}e^{\nu,2}(\partial_{x}e^{1}_{\nu}-\partial_{\nu}e^{1}_{x})-\frac{1}{2}e^{\rho,a}e^{\sigma,b}(\partial_{\rho}e_{\sigma,c}-\partial_{\sigma}e_{\rho,c})e^{c}_{x};$
$\displaystyle\omega_{y}^{12}=\frac{1}{2}e^{\nu,1}(\partial_{y}e^{2}_{\nu}-\partial_{\nu}e^{2}_{y})-\frac{1}{2}e^{\nu,2}(\partial_{y}e^{1}_{\nu}-\partial_{\nu}e^{1}_{y})-\frac{1}{2}e^{\rho,a}e^{\sigma,b}(\partial_{\rho}e_{\sigma,c}-\partial_{\sigma}e_{\rho,c})e^{c}_{y}$

These equations are further simplified with the help of equations $(11-17)$
with $e^{1}_{y}=0$ , $e^{1}_{x}=1$ and the Burger tensor
$\partial_{x}e^{2}_{y}-\partial_{y}e^{2}_{x}=B^{(2)}\delta^{2}(\vec{r})$ .

$\displaystyle\omega_{x}^{12}=\frac{1}{2}g^{\nu,\lambda}e^{1}_{\lambda}(\partial_{x}e^{2}_{\nu}-\partial_{\nu}e^{2}_{x})-\frac{1}{2}g^{\rho,r}e^{1}_{r}g^{\rho,s}e^{2}_{s}[\partial_{\rho}(\delta_{c,b}e^{b}_{\sigma})-\partial_{\sigma}(\delta_{c,d}e^{d}_{\rho})]e^{c}_{x}=$
$\displaystyle\frac{1}{2}B^{(2)}\delta^{(2)}(\vec{r})[g^{y,x}e^{1}_{x}+g^{y,y}e^{1}_{y}-(g^{x,r}g^{y,s}-g^{y,r}g^{x,s})(e^{1}_{r}e^{2}_{s}e^{2}_{x})]=$
$\displaystyle\frac{1}{2}B^{(2)}\delta^{(2)}(\vec{r})[g^{y,x}e^{1}_{x}-(g^{x,x}g^{y,y}-g^{y,x}g^{x,y})e^{1}_{x}e^{2}_{y}e^{2}_{x}]\approx$
$\displaystyle\frac{1}{2}B^{(2)}\delta^{(2)}(\vec{r})[-\frac{B^{(2)}}{2\pi}\frac{y}{y^{2}+x^{2}}-(1-(\frac{B^{(2)}}{2\pi}\frac{y}{y^{2}+x^{2}})^{2})(\frac{B^{(2)}}{2\pi}\frac{y}{x^{2}+y^{2}})(1-\frac{B^{(2)}}{2\pi}\frac{x}{x^{2}+y^{2}})]\approx$
$\displaystyle\frac{1}{2}B^{(2)}\delta^{(2)}(\vec{r})[-\frac{B^{(2)}}{2\pi}\frac{2y-x}{y^{2}+x^{2}}]$

and

$\displaystyle\omega_{y}^{12}=\frac{1}{2}e^{\nu,1}(\partial_{y}e^{2}_{\nu}-\partial_{\nu}e^{2}_{y})-\frac{1}{2}e^{\nu,2}(\partial_{y}e^{1}_{\nu}-\partial_{\nu}e^{1}_{y})-\frac{1}{2}e^{\rho,1}e^{\sigma,2}[\partial_{\rho}(\delta_{c,b}e_{\sigma}^{b})-\partial_{\sigma}(\delta_{c,d}e_{\rho}^{d})]e^{c}_{y}=$
$\displaystyle\frac{1}{2}g^{\nu,\lambda}e^{1}_{\lambda}[\partial_{y}e^{2}_{\nu}-\partial_{\nu}e^{2}_{y}]-\frac{1}{2}g^{\nu,r}e^{1}_{r}[\partial_{y}e^{1}_{\nu}-\partial_{\nu}e^{1}_{y}]-\frac{1}{2}g^{\rho,r}e^{1}_{r}g^{\sigma,s}e^{2}_{s}[\partial_{\rho}e^{c}_{\sigma}-\partial_{\sigma}e^{c}_{\rho}]e^{c}_{y}=$
$\displaystyle-\frac{B^{(2)}}{2}\delta^{(2)}(\vec{r})g^{x,\lambda}e^{1}_{\lambda}-\frac{B^{(2)}}{2}\delta^{(2)}(\vec{r})[g^{x,r}g^{y,s}-g^{y,r}g^{x,s}]e^{1}_{r}e^{2}_{s}e^{2}_{y}\approx-\frac{B^{(2)}}{2}\delta^{(2)}(\vec{r})$

To first order first the Burger vector $B^{(2)}$ the spin connections are
given by : $\omega_{x}^{12}=-\omega_{x}^{21}\approx 0$ and
$\omega_{y}^{12}=-\omega_{y}^{21}\approx-\frac{1}{2}B^{(2)}\delta^{2}(\vec{r})$.

Figure 1: The contours
$(x(s))^{2}+(y(s)-\frac{R_{g}}{n})^{2}=(\frac{R_{g}}{n})^{2}$ for $n=\pm 1,\pm
2,\pm 3$(in decreasing size ),$R_{g}(n)=\frac{R_{g}}{n}$. $n=0$ corresponds to
the equation $y(s)=0$ and $|y|>d$ (see the text). The the distance is measured
in units of the Burger vector $B^{(2)}$.

Figure 2: The tunneling density of states for $n=0$ , $\frac{dI}{dV}\propto
D^{(n=0)}(\frac{x}{B^{(2)}},\frac{y}{B^{(2)}};\mu=120mV)$. The right corner
represents the intersection of the $x$ coordinate which runs from $30$ (right
corner) to $-30$ and the $y$ coordinate which runs from $-30$ (right corner)
to $30$ in units of the Burger vector.

Figure 3: The tunneling density of states for $n=0$ as a function of $y$ and
$V$ $\frac{dI}{dV}\propto
D^{(n=0)}(\frac{x}{B^{(2)}}=-2,\frac{y}{B^{(2)}};\mu=120mV)$. The voltage
range is $-120\leq V\leq 50$ and the $y$ coordinate is in the range
$-30\leq\frac{y}{B^{(2)}}\leq 30$.

Figure 4: Many Dislocations - with the core of the dislocations at
$[x_{w},y=0]$ , $w=1,2...2M$; The maximum of the tunneling density of states
is confined along $y=0$. The coordinates of the tunneling density of states
are restricted to : $-40\leq\frac{x}{B^{(2)}}\leq 40$ and
$-20\leq\frac{y}{B^{(2)}}\leq 20$.

Figure 5: The discrete tunneling density of states for $n=1$, as a function of
the voltage $V$
$D^{(n=1)}(V;\theta=\frac{\pi}{2},\frac{u}{B^{(2)}},\mu=120mV)$

Figure 6: The tunneling density of states as a function of $\theta$
$D^{(n=1)}(\theta;\frac{u}{B^{(2)}}=0.01,V=280mV,\mu=120mV)$

Figure 7: The tunneling density of states as a function of $\theta$ and $u$ at
a fixed voltage $V=280mV$
$D^{(n=1)}(\theta,\frac{u}{B^{(2)}};V=280mV,\mu=120mV)$

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