**Abstract**

An important measurable quantity in the carbon nanostructures, including the nanotubular part of the graphitic wormhole, is the spin-orbit coupling. We will present in this chapter spin-orbit coupling for the fermions located in exotic graphene structures as is graphene wormhole and also in biological systems. Considering this influence, the two-component Dirac equation is changed into the usual four-component form. As a consequence, the chiral fermions should be detected close to the wormhole bridge. We will show that the smaller is the radius of the wormhole bridge, the stronger this effect should be. Finally, we will describe the role of spinor fields in the time series of genetic code. The reversed transcription process of the gene expression could be defined by a moduli state space model of a coupling spinor field between the gene of a viral particle and the host cell. As a general result, all states of codon can be computed by the Chern-Simons 3-forms.

**Keywords:** spinor network structure, spin orbit coupling, Chern-Simons fields, graphene wormhole, genetic code

## **1. Graphitic wormhole**

The investigation of unique chemical and mechanical properties of nanostructures, e.g., fullerene, graphene, and nanotubes, promises a wide application in many technical areas. The electronic properties of the nanostructures are basically defined by their hexagonal carbon lattice structure and its variations. New promising results are expected with the preparation of more complicated forms as a wormhole. The wormhole is usually composed of two different kinds of nanostructure: two graphene sheets are connected together with the help of a connecting nanotube [1] (**Figure 1**). This is achieved by a supply of two sets of six heptagonal defects onto both sides of the given nanotube. There exists the restrictions on the form of the nanotube—the chirality must be 6ð Þ *n;* 6*n* armchair or 6ð Þ *n;* 0 zigzag and a radius of the nanotube is larger than its length.

The metric tensor of the wormhole is given by

$$\mathbf{g}\_{\mu\nu} = \Lambda^2(r\_{\pm}) \begin{pmatrix} \mathbf{1} & \mathbf{0} \\ \mathbf{0} & r\_{\pm}^2 \end{pmatrix}, \quad \Lambda(r\_{\pm}) = (\mathfrak{a}/r\_{\pm})^2 \theta(\mathfrak{a} - r\_{\pm}) + \theta(r\_{\pm} - \mathfrak{a}), \tag{1}$$

<sup>Ω</sup>*<sup>φ</sup>* ¼ � <sup>i</sup> 2 *σ*<sup>3</sup> *r* Λ0 ð Þ*r* Λð Þ*r*

*DOI: http://dx.doi.org/10.5772/intechopen.88486*

and after the substitution into Eq. (2), we get the equation

*Application of Spin-Orbit Coupling in Exotic Graphene Structures and Biology*

where each sign corresponds to a different Dirac point

�i*vF <sup>∂</sup><sup>r</sup>* <sup>þ</sup>

�i*vF <sup>∂</sup><sup>r</sup>* � <sup>1</sup>

i*vF <sup>r</sup> a* � �<sup>2</sup> *<sup>∂</sup><sup>r</sup>* � <sup>1</sup>

i*vF <sup>r</sup> a* � �<sup>2</sup> *<sup>∂</sup><sup>r</sup>* <sup>þ</sup>

for 0< *r*≤*a*. For *r*≥*a*, the solution is

*<sup>ψ</sup>*� <sup>¼</sup> *<sup>ψ</sup>*�

þ *c*<sup>2</sup>

for *r*≥*a* and

component *ψ*�

for *r*≥*a* and

for *r*≥*a* and

*ψ*<sup>þ</sup>

**97**

for 0< *r*≤*a*. For *ψ*�

*<sup>B</sup>* . For the value *<sup>a</sup><sup>φ</sup>* <sup>¼</sup> <sup>1</sup>

2

<sup>i</sup>*vFσ<sup>μ</sup> <sup>∂</sup><sup>μ</sup>* <sup>þ</sup> <sup>Ω</sup>*μ*∓<sup>i</sup> *<sup>a</sup><sup>μ</sup>*

1 *r* i*∂θ*∓ *aφ r* þ 1 2*r*

*r* <sup>i</sup>*∂<sup>θ</sup>* � *aφ r* þ 1 2*r*

*r* <sup>i</sup>*∂<sup>θ</sup>* � *aφ <sup>r</sup>* � <sup>1</sup> 2*r*

1 *r* i*∂θ*∓ *aφ <sup>r</sup>* � <sup>1</sup> 2*r*

*<sup>A</sup>*ð Þ *r; φ*

*<sup>∂</sup><sup>r</sup>* � <sup>1</sup> *r* i*∂θ*∓ *aφ r* þ 1 2*r*

*<sup>∂</sup><sup>r</sup>* � <sup>1</sup> *r* <sup>i</sup>*∂<sup>θ</sup>* � *aφ <sup>r</sup>* � <sup>1</sup> 2*r*

*<sup>B</sup>* and the value *<sup>a</sup><sup>φ</sup>* <sup>¼</sup> <sup>3</sup>

*ψ*�

*ψ*�

!

*ψ*� *<sup>B</sup>* ð Þ *r; φ*

� �*ψ*�

� �*ψ*�

� �*ψ*�

� �*ψ*�

¼ *c*<sup>1</sup>

*Yn*∓*aφ*�1*=*2ð Þ *kr* �i sgn *<sup>ε</sup>Yn*∓*aφ*þ1*=*2ð Þ *kr* !*,*

*<sup>A</sup>* of the solution to be equal to zero, one get from (6) and (7)

where the energy *ε* ¼ �*vFk*, *Jn*ð Þ *x* and *Yn*ð Þ *x* are the Bessel functions. The zero modes solve the Dirac equation for zero energy. If one choose the

� �*ψ*�

� �*ψ*�

*<sup>B</sup>* ð Þ� *r; φ r*

*<sup>B</sup>* ð Þ� *r; φ r*

þ 1

� �*,* <sup>Ω</sup>*<sup>r</sup>* <sup>¼</sup> <sup>0</sup>*,* (4)

� �*ψ*� <sup>¼</sup> *εψ*�*,* (5)

*<sup>B</sup>* ¼ *εψ*� *A,*

*<sup>A</sup>* ¼ *εψ*� *B*

*<sup>B</sup>* ¼ *εψ*� *A,*

*<sup>A</sup>* ¼ *εψ*� *B*

*Jn*∓*aφ*�1*=*2ð Þ *kr* �i sgn *<sup>ε</sup>Jn*∓*aφ*þ1*=*2ð Þ *kr* !

<sup>2</sup>, the solution is

, possible solutions are not strictly normalizable, and the

�*n*�2 *e*

�*n*þ2 *e*

for 0< *r*≤*a*. For both cases, it is normalizable only for *n* ¼ 0, and so this is the only solution. In a similar way, we can calculate the zero modes for the component

*<sup>B</sup>* ¼ 0 (9)

*<sup>B</sup>* ¼ 0 (10)

<sup>i</sup> *<sup>n</sup><sup>φ</sup>* (11)

<sup>i</sup> *<sup>n</sup><sup>φ</sup>* (12)

(6)

(7)

(8)

**Figure 1.**

*Schematic representation of graphitic wormhole consisting from two graphene sheets connected together with the help of a nanotube.*

where *θ* is the Heaviside step function, *r*� and *r*<sup>þ</sup> are the polar coordinates of lower and upper graphene sheets, and *a* ¼ ffiffiffiffiffiffiffiffiffiffi *r*�*r*<sup>þ</sup> p is the radius of the wormhole.

#### **1.1 Electronic structure**

We consider the continuum gauge field theory, i.e., at each point of a molecular surface, we take into account an influence of different gauge fields that enter the Dirac-like equation for an electron

$$\mathrm{i}\nu\_{F}\sigma^{\mu}\left[\partial\_{\mu}+\Omega\_{\mu}-\mathrm{i}\mathfrak{a}\_{\mu}-\mathrm{i}\mathfrak{a}\_{\mu}^{W}-\mathrm{i}\mathcal{A}\_{\mu}\right]\psi=\mathrm{E}\psi,\tag{2}$$

with *σ<sup>α</sup>* matrices as the Pauli matrices, the Fermi velocity *vF*, the spin connection

$$
\Omega\_{\mu} = \frac{1}{8} \alpha\_{\mu}^{a\beta} \left[ \sigma\_a, \sigma\_\beta \right], \tag{3}
$$

and the covariant derivative <sup>∇</sup>*<sup>μ</sup>* <sup>¼</sup> *<sup>∂</sup><sup>μ</sup>* <sup>þ</sup> <sup>Ω</sup>*μ*. The gauge fields *<sup>a</sup>μ, aW <sup>μ</sup>* are caused by the presence of the defects, and by rotational symmetry, the gauge field *A<sup>μ</sup>* characterizes the possible magnetic field.

In the case of the wormhole with the metric Eq. (1), the effective flux caused by the presence of the defects is included in the gauge field *aμ*, and for the particular polar components, it has the values *<sup>a</sup><sup>φ</sup>* <sup>¼</sup> <sup>3</sup> <sup>2</sup>, *ar* ¼ 0 for two possibilities: the first corresponds to the chiral vector with the form 6ð Þ *n;* 6*n* , the second corresponds to the chiral vector with the form 6ð Þ *n;* 0 and *n* divisible by 3. In the case of chiral vector of the form 6ð Þ *n;* 0 , where *n* is not divisible by 3, the components of the gauge field are *<sup>a</sup><sup>φ</sup>* <sup>¼</sup> <sup>1</sup> 2 , *ar* ¼ 0. Regarding that the components of the spin connection are

*Application of Spin-Orbit Coupling in Exotic Graphene Structures and Biology DOI: http://dx.doi.org/10.5772/intechopen.88486*

$$\boldsymbol{\Omega}\_{\boldsymbol{\varphi}} = -\frac{\mathrm{i}}{2} \sigma\_3 \left( r \frac{\boldsymbol{\Lambda}'(r)}{\boldsymbol{\Lambda}(r)} + \mathbf{1} \right), \quad \boldsymbol{\Omega}\_r = \mathbf{0}, \tag{4}$$

and after the substitution into Eq. (2), we get the equation

$$\mathrm{i}\nu\_{F}\sigma^{\mu}\left(\partial\_{\mu}+\Omega\_{\mu}\mp\mathrm{i}\,\mathfrak{a}\_{\mu}\right)\psi^{\pm}=\epsilon\psi^{\pm},\tag{5}$$

where each sign corresponds to a different Dirac point

$$\begin{split} -\mathrm{i}\nu\_{F} \left( \partial\_{r} + \frac{\mathbf{1}}{r} \mathrm{i}\partial\_{\theta} \mp \frac{a\_{\varphi}}{r} + \frac{\mathbf{1}}{2r} \right) \mathsf{y}\_{B}^{\pm} &= \varepsilon \mathsf{y}\_{A}^{\pm}, \\ -\mathrm{i}\nu\_{F} \left( \partial\_{r} - \frac{\mathbf{1}}{r} \mathrm{i}\partial\_{\theta} \pm \frac{a\_{\varphi}}{r} + \frac{\mathbf{1}}{2r} \right) \mathsf{y}\_{A}^{\pm} &= \varepsilon \mathsf{y}\_{B}^{\pm} \end{split} \tag{6}$$

for *r*≥*a* and

where *θ* is the Heaviside step function, *r*� and *r*<sup>þ</sup> are the polar coordinates of

*Schematic representation of graphitic wormhole consisting from two graphene sheets connected together with the*

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

We consider the continuum gauge field theory, i.e., at each point of a molecular surface, we take into account an influence of different gauge fields that enter the

with *σ<sup>α</sup>* matrices as the Pauli matrices, the Fermi velocity *vF*, the spin connection

the presence of the defects, and by rotational symmetry, the gauge field *A<sup>μ</sup>* charac-

In the case of the wormhole with the metric Eq. (1), the effective flux caused by the presence of the defects is included in the gauge field *aμ*, and for the particular

corresponds to the chiral vector with the form 6ð Þ *n;* 6*n* , the second corresponds to the chiral vector with the form 6ð Þ *n;* 0 and *n* divisible by 3. In the case of chiral vector of the form 6ð Þ *n;* 0 , where *n* is not divisible by 3, the components of the

, *ar* ¼ 0. Regarding that the components of the spin

h i

<sup>i</sup>*vFσ<sup>μ</sup> <sup>∂</sup><sup>μ</sup>* <sup>þ</sup> <sup>Ω</sup>*<sup>μ</sup>* � *ia<sup>μ</sup>* � *ia<sup>W</sup>*

<sup>Ω</sup>*<sup>μ</sup>* <sup>¼</sup> <sup>1</sup> 8 *ωαβ <sup>μ</sup> σα; σβ*

and the covariant derivative <sup>∇</sup>*<sup>μ</sup>* <sup>¼</sup> *<sup>∂</sup><sup>μ</sup>* <sup>þ</sup> <sup>Ω</sup>*μ*. The gauge fields *<sup>a</sup>μ, aW*

*r*�*r*<sup>þ</sup>

*<sup>μ</sup>* � *iA<sup>μ</sup>*

p is the radius of the wormhole.

*ψ* ¼ *Eψ,* (2)

*<sup>μ</sup>* are caused by

� �*,* (3)

<sup>2</sup>, *ar* ¼ 0 for two possibilities: the first

lower and upper graphene sheets, and *a* ¼ ffiffiffiffiffiffiffiffiffiffi

**1.1 Electronic structure**

**Figure 1.**

*help of a nanotube.*

Dirac-like equation for an electron

terizes the possible magnetic field.

gauge field are *<sup>a</sup><sup>φ</sup>* <sup>¼</sup> <sup>1</sup>

connection are

**96**

polar components, it has the values *<sup>a</sup><sup>φ</sup>* <sup>¼</sup> <sup>3</sup>

2

$$\begin{split} \dot{\nu}\nu\_{F}\left(\frac{r}{a}\right)^{2} \left(\partial\_{r} - \frac{\mathbf{1}}{r} \dot{\imath}\partial\_{\theta} \pm \frac{a\_{\theta}}{r} - \frac{\mathbf{1}}{2r}\right) \nu\_{B}^{\pm} &= \epsilon \nu\_{A}^{\pm}, \\ \dot{\nu}\_{F}\left(\frac{r}{a}\right)^{2} \left(\partial\_{r} + \frac{\mathbf{1}}{r} \dot{\imath}\partial\_{\theta} \mp \frac{a\_{\theta}}{r} - \frac{\mathbf{1}}{2r}\right) \nu\_{A}^{\pm} &= \epsilon \nu\_{B}^{\pm} \end{split} \tag{7}$$

for 0< *r*≤*a*. For *r*≥*a*, the solution is

$$\begin{split} \boldsymbol{y}^{\pm} = \begin{pmatrix} \boldsymbol{y}\_{A}^{\pm}(\boldsymbol{r}, \boldsymbol{\varrho}) \\ \boldsymbol{y}\_{B}^{\pm}(\boldsymbol{r}, \boldsymbol{\varrho}) \end{pmatrix} = \boldsymbol{c}\_{1} \begin{pmatrix} \boldsymbol{J}\_{\boldsymbol{n} \mp \boldsymbol{a}\_{\theta} - 1/2}(\boldsymbol{k} \boldsymbol{r}) \\ -\mathbf{i} \text{ sgn } \boldsymbol{\varepsilon} \boldsymbol{I}\_{\boldsymbol{n} \mp \boldsymbol{a}\_{\theta} + 1/2}(\boldsymbol{k} \boldsymbol{r}) \end{pmatrix} \\ + \boldsymbol{c}\_{2} \begin{pmatrix} \boldsymbol{Y}\_{\boldsymbol{n} \mp \boldsymbol{a}\_{\psi} - 1/2}(\boldsymbol{k} \boldsymbol{r}) \\ -\mathbf{i} \text{ sgn } \boldsymbol{\varepsilon} \boldsymbol{Y}\_{\boldsymbol{n} \mp \boldsymbol{a}\_{\psi} + 1/2}(\boldsymbol{k} \boldsymbol{r}) \end{pmatrix}, \end{split} \tag{8}$$

where the energy *ε* ¼ �*vFk*, *Jn*ð Þ *x* and *Yn*ð Þ *x* are the Bessel functions.

The zero modes solve the Dirac equation for zero energy. If one choose the component *ψ*� *<sup>A</sup>* of the solution to be equal to zero, one get from (6) and (7)

$$(\partial\_r - \frac{1}{r} \mathbf{i} \partial\_\theta \mp \frac{a\_\varphi}{r} + \frac{1}{2r}) \boldsymbol{\mu}\_B^\pm = \mathbf{0} \tag{9}$$

for *r*≥*a* and

$$(\partial\_r - \frac{1}{r} \mathbf{i} \partial\_\theta \pm \frac{a\_\varphi}{r} - \frac{1}{2r}) \varphi\_B^{\pm} = \mathbf{0} \tag{10}$$

for 0< *r*≤*a*. For *ψ*� *<sup>B</sup>* and the value *<sup>a</sup><sup>φ</sup>* <sup>¼</sup> <sup>3</sup> <sup>2</sup>, the solution is

$$
\psi\_B^-(r,\rho) \sim r^{-n-2} e^{i n \rho} \tag{11}
$$

for *r*≥*a* and

$$
\psi\_B^-(r,\rho) \sim r^{-n+2} e^{in\rho} \tag{12}
$$

for 0< *r*≤*a*. For both cases, it is normalizable only for *n* ¼ 0, and so this is the only solution. In a similar way, we can calculate the zero modes for the component *ψ*<sup>þ</sup> *<sup>B</sup>* . For the value *<sup>a</sup><sup>φ</sup>* <sup>¼</sup> <sup>1</sup> 2 , possible solutions are not strictly normalizable, and the

zero modes exist only for the case of the connecting nanotube being armchair or zigzag with the chiral vector 6ð Þ *n;* 0 , *n* divisible by 3. In other cases the zero modes do not exist.

with *R* as the radius of the cylinder. The solution of this equation has the form

A similar form has the dispersion relation associated with the massive 1D Dirac

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>M</sup>*<sup>2</sup> � *<sup>E</sup>*<sup>2</sup> <sup>p</sup>

where *M* is the mass of corresponding fermion. It is proven [9] that for a suitable choice of the parameters, 2D massless case is in analogy with 1D massive case, and

*gξξ gφφ* <sup>þ</sup> *<sup>E</sup>*<sup>2</sup> � *<sup>M</sup>*<sup>2</sup> � �*gξξ* !*uj* <sup>¼</sup> <sup>0</sup>*,* (18)

where *M* is the mass of the corresponding fermion. For different values of *M*, one can find the corrections of local density of states for the graphitic wormhole. It seems that these massive particles arising in the wormhole nanotubes could create energy bulks on and near the wormhole bridge which should be experimentally measured by the STM or Raman spectroscopy [10]. Another possibly identification of wormhole structure comes from the fact of creation of strain solitons and topological defects by massive particles on the bridge of bilayer graphene which should propagate throughout the graphene sheet. These are almost macroscopic effects

An important measurable quantity in carbon nanostructures, which includes a nanotubular part of a graphitic wormhole, is a spin-orbit coupling (SOC) [12, 13]. If one considers this influence, two-component Dirac equation could be changed into the usual four-component form, and as a consequence chiral fermions should

One can reflect on two sources of SOC: (1) the interatomic one that preserves the *z*-component of a spin and (2) so-called Rashba-type coming from the external electric field, which conserves the *z*-component of an angular momentum *Jz*. In both cases, the strength of SOC is influenced by the nonzero curvature. In the next

Considering the SOC we can write the Dirac equation for the nanotube in the

*, F<sup>K</sup>*

*A FK B*

*<sup>B</sup>* <sup>¼</sup> *<sup>F</sup><sup>K</sup>*

¼ *E*

*B,*↑ *FK B,*↓

!

*FK A FK B*

*,* (19)

*:* (20)

!

!

<sup>¼</sup> <sup>0</sup> ^*<sup>f</sup>* ^*f* † 0 ! *<sup>F</sup><sup>K</sup>*

> *A,*↑ *FK A,*↓

!

*, k* ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~*j*2 *<sup>R</sup>*<sup>2</sup> � *<sup>E</sup>*<sup>2</sup> *:*

*,* (17)

(16)

s

*uj*ð Þ¼ *<sup>ξ</sup> Aek<sup>ξ</sup>* <sup>þ</sup> *Be*�*k<sup>ξ</sup>*

*Application of Spin-Orbit Coupling in Exotic Graphene Structures and Biology*

*k* ¼

*<sup>∂</sup>ξgφφ* �~*j*<sup>2</sup>

ffiffiffiffiffiffiffi *gξξ g*3 *φφ* s

and should be caught by the experimental physicists [11].

**1.3 Spin-orbit coupling in the wormhole connecting nanotube**

equation

form

**99**

where

one can rewrite Eq. (13) in the form

*DOI: http://dx.doi.org/10.5772/intechopen.88486*

*<sup>∂</sup>ξgξξ* <sup>þ</sup> <sup>~</sup>*<sup>j</sup>* 2

be detected close to the wormhole bridge.

we will be interested in the first source of the SOC.

!

*FK*

*<sup>A</sup>* <sup>¼</sup> *<sup>F</sup><sup>K</sup>*

*<sup>H</sup>*^ *<sup>F</sup><sup>K</sup> A FK B*

*<sup>∂</sup>ξξ* � <sup>1</sup> 2*gξξ*

Recently, in work [2] some peculiarities in the bilayer graphene were analytically predicted. A possible indication of the wormhole could be found in [3, 4], where a new type of zero modes is investigated. These zero modes could be the zero modes studied in this subsection applied to the case of the smallest wormhole.

#### **1.2 Case of massive fermions**

Up to now we supposed that the fermions appearing in the Dirac equation have the zero mass or that the mass is very small in comparison with their energy, but in [5, 6] it was shown that the Fermi velocity needs to be renormalized due to the elasticity and deformations in a graphene. In our case of the graphitic wormhole, including big deformations, the velocity of fermions close to the wormhole bridge could achieve such values that the relativistic effects can appear or break off the symmetry [7] and the mass of fermions would be non-negligible. The radius of the wormhole and its bridge is very small in comparison with the size of the upper and the lower graphene sheet (**Figure 2**) and by folding the sheet into a tube they acquire nonzero effective mass as they move along the tube axis. This change of the space topology of graphene from 2D to 1D space compactification is similar to the string theory compactification, and we can imagine a wormhole connecting nanotubes as 1D object.

To include the mass into the Dirac Eq. (2), one can transform the system of equations [8] into the differential equation of the second order

$$
\left(\partial\_{\xi\xi} - \frac{1}{2\mathfrak{g}\_{\xi\xi}}\partial\_{\xi}\mathfrak{g}\_{\xi\xi} + \frac{\tilde{j}}{2}\sqrt{\frac{\mathfrak{g}\_{\xi\xi}}{\mathfrak{g}\_{\varrho\eta}^{3}}}\partial\_{\xi}\mathfrak{g}\_{\varrho\eta} - \tilde{j}2\frac{\mathfrak{g}\_{\xi\xi}}{\mathfrak{g}\_{\varrho\eta}} + E^{2}\mathfrak{g}\_{\xi\xi}\right)u\_{j} = 0.\tag{13}
$$

One can suppose cylindrical geometry in order to simplify the equation into

$$\left(\partial\_{\xi\xi} + E^2 - \frac{\tilde{j}\,2}{R^2}\right)u\_{\tilde{j}} = 0,\tag{14}$$

if a radius vector of the point at the surface will have the form

$$\overrightarrow{R} = (R\cos\varphi, R\sin\varphi, \xi),\tag{15}$$

**Figure 2.** *The simplest realization of smallest graphitic wormholes.*

*Application of Spin-Orbit Coupling in Exotic Graphene Structures and Biology DOI: http://dx.doi.org/10.5772/intechopen.88486*

with *R* as the radius of the cylinder. The solution of this equation has the form

$$u\_j(\xi) = Ae^{k\xi} + Be^{-k\xi}, \quad k = \sqrt{\frac{\tilde{j}\,2}{R^2} - E^2}. \tag{16}$$

A similar form has the dispersion relation associated with the massive 1D Dirac equation

$$k = \sqrt{{\bf M}^2 - E^2},\tag{17}$$

where *M* is the mass of corresponding fermion. It is proven [9] that for a suitable choice of the parameters, 2D massless case is in analogy with 1D massive case, and one can rewrite Eq. (13) in the form

$$
\left(\partial\_{\xi\xi} - \frac{1}{2\mathfrak{g}\_{\xi\xi}}\partial\_{\xi}\mathfrak{g}\_{\xi\xi} + \frac{\tilde{j}}{2}\sqrt{\frac{\mathfrak{g}\_{\xi\xi}}{\mathfrak{g}\_{\eta\eta}^{3}}}\partial\_{\xi}\mathfrak{g}\_{\eta\eta} - \tilde{j}2\frac{\mathfrak{g}\_{\xi\xi}}{\mathfrak{g}\_{\eta\eta}} + \left(E^{2} - M^{2}\right)\mathfrak{g}\_{\xi\xi}\right)\mathfrak{u}\_{j} = 0,\tag{18}
$$

where *M* is the mass of the corresponding fermion. For different values of *M*, one can find the corrections of local density of states for the graphitic wormhole. It seems that these massive particles arising in the wormhole nanotubes could create energy bulks on and near the wormhole bridge which should be experimentally measured by the STM or Raman spectroscopy [10]. Another possibly identification of wormhole structure comes from the fact of creation of strain solitons and topological defects by massive particles on the bridge of bilayer graphene which should propagate throughout the graphene sheet. These are almost macroscopic effects and should be caught by the experimental physicists [11].

#### **1.3 Spin-orbit coupling in the wormhole connecting nanotube**

An important measurable quantity in carbon nanostructures, which includes a nanotubular part of a graphitic wormhole, is a spin-orbit coupling (SOC) [12, 13]. If one considers this influence, two-component Dirac equation could be changed into the usual four-component form, and as a consequence chiral fermions should be detected close to the wormhole bridge.

One can reflect on two sources of SOC: (1) the interatomic one that preserves the *z*-component of a spin and (2) so-called Rashba-type coming from the external electric field, which conserves the *z*-component of an angular momentum *Jz*. In both cases, the strength of SOC is influenced by the nonzero curvature. In the next we will be interested in the first source of the SOC.

Considering the SOC we can write the Dirac equation for the nanotube in the form

$$
\hat{H} \begin{pmatrix} F\_A^K \\ F\_B^K \end{pmatrix} = \begin{pmatrix} \mathbf{0} & \hat{f} \\ \hat{f}^\dagger & \mathbf{0} \end{pmatrix} \begin{pmatrix} F\_A^K \\ F\_B^K \end{pmatrix} = E \begin{pmatrix} F\_A^K \\ F\_B^K \end{pmatrix}, \tag{19}
$$

where

$$F\_A^K = \begin{pmatrix} F\_{A,\uparrow}^K \uparrow \\ F\_{A,\downarrow}^K \end{pmatrix}, \quad F\_B^K = \begin{pmatrix} F\_{B,\uparrow}^K \uparrow \\ F\_{B,\downarrow}^K \end{pmatrix}. \tag{20}$$

zero modes exist only for the case of the connecting nanotube being armchair or zigzag with the chiral vector 6ð Þ *n;* 0 , *n* divisible by 3. In other cases the zero modes

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

Recently, in work [2] some peculiarities in the bilayer graphene were analytically predicted. A possible indication of the wormhole could be found in [3, 4], where a new type of zero modes is investigated. These zero modes could be the zero modes studied in this subsection applied to the case of the smallest wormhole.

Up to now we supposed that the fermions appearing in the Dirac equation have the zero mass or that the mass is very small in comparison with their energy, but in [5, 6] it was shown that the Fermi velocity needs to be renormalized due to the elasticity and deformations in a graphene. In our case of the graphitic wormhole, including big deformations, the velocity of fermions close to the wormhole bridge could achieve such values that the relativistic effects can appear or break off the symmetry [7] and the mass of fermions would be non-negligible. The radius of the wormhole and its bridge is very small in comparison with the size of the upper and the lower graphene sheet (**Figure 2**) and by folding the sheet into a tube they acquire nonzero effective mass as they move along the tube axis. This change of the space topology of graphene from 2D to 1D space compactification is similar to the string theory compactification, and we can imagine a wormhole connecting

To include the mass into the Dirac Eq. (2), one can transform the system of

One can suppose cylindrical geometry in order to simplify the equation into

*R*2

*<sup>∂</sup>ξgφφ* �~*j*<sup>2</sup>

*gξξ gφφ*

<sup>þ</sup> *<sup>E</sup>*<sup>2</sup> *gξξ*

¼ ð Þ *R* cos *φ; R* sin *φ; ξ ,* (15)

*uj* ¼ 0*,* (14)

*uj* ¼ 0*:* (13)

ffiffiffiffiffiffiffi *gξξ g*3 *φφ*

*<sup>∂</sup>ξξ* <sup>þ</sup> *<sup>E</sup>*<sup>2</sup> � <sup>~</sup>*j*<sup>2</sup>

if a radius vector of the point at the surface will have the form

*R* !

� �

!

s

equations [8] into the differential equation of the second order

*<sup>∂</sup>ξgξξ* <sup>þ</sup> <sup>~</sup>*<sup>j</sup>* 2

do not exist.

**1.2 Case of massive fermions**

nanotubes as 1D object.

**Figure 2.**

**98**

*<sup>∂</sup>ξξ* � <sup>1</sup> 2*gξξ*

*The simplest realization of smallest graphitic wormholes.*

The expression ^*f* has the following form

$$\hat{f} = \chi \left(\hat{k}\_{\text{x}} - i\hat{k}\_{\text{y}}\right) + i\frac{\delta\gamma'}{4R}\hat{\sigma}\_{\text{x}}\left(\overrightarrow{r}\right) - \frac{2\delta\gamma p}{R}\hat{\sigma}\_{\text{y}}\tag{21}$$

the geometry of the corresponding graphene sheets will be curved, and this brings a

*Application of Spin-Orbit Coupling in Exotic Graphene Structures and Biology*

The effects connected with the deformation of a graphene and a consequent change of the distance of the carbon atoms in the layer are described in [14]. It causes the rotation of the *pz* orbitals and rehybridization of the *π* and *σ* orbitals. The procedure leads to the creation of the *p n* junctions similarly to the case of a transistor. This effect changes the Fermi level which is rising in the far areas from the wormhole center. The electron flux is directed from these areas to the middle where the electric charge is accumulated, and in the case of the deformed wormhole, one can speak about so-called graphene black hole. The form of a middle part of the nanotube plays a big role for this purpose. It cannot be unperturbed because in such a case the effect of the black hole would be disrupted. It can be ensured only in the case when the nanotubular neck is tapering in the direction to its center, because this ensures the decrease of the Fermi level [15]. The related effects which

appear on the nanostructures are also described in [16], where the special

the defects with the applications in cosmological models.

**2. Spinor fields in biological systems**

code of the protein curvature.

**101**

string theory and M- and G-theories [21, 22].

relativistic-like properties of the Beltrami pseudosphere naturally point to quantum field theory in curved space. In the work the finite temperature local density of states is predicted that is a realization of the Hawking-Unruh effect. Mentioned effect of the graphene black hole could eventually disappear in the presence of external magnetic (electric) field which would cause the transfer of the charge from one wormhole sheet to another one through a nanotube center. This serves as an important model for further investigations of the electron flux in the presence of

One of the present problems in genetic engineering is the prediction of biological

Today, a genetical structure is studied by standard alphabet codes *A*, *T*, *C*, *G*, and *U* as a sequence of strings for the representation of genetic code for various organisms without any exact definition of a new time series of genetic code [17] in contrast to standard time series modeling. With this representation [18], it is very difficult to calculate the genetic variation [19] and to perform calculations within a framework of self-consistent mathematical theory [20], namely, in the context of

There are still attempts to perform empirical data analysis of the genetic varia-

tion [23] and to detect the pattern matching over the gene sequence by using algorithm over a standard alphabet code as their time series representation. It seems

gene variation and the representation of corresponding genetic code. This issue emerges in the plotting graphs related to the connection curvature of a docking processes. The docking process is important in the genes of the protein structure and could be adopted instead of using a very long alphabet notation as the string sequence and the comparison of the sequences of docking. From this point of view, methods of quantum field theory, general relativity, and related tools can be of high interest. The equilibrium between the supersymmetry and the mirror symmetry of the left-handed and right-handed DNA, RNA, nucleic and amino acid molecules can be explained by anti-de Sitter (AdS) correspondence in the Yang-Mills theory and the Chern-Simon currents in biology as the curvature of the spectrum in genetic

significant change of the physical properties.

*DOI: http://dx.doi.org/10.5772/intechopen.88486*

**1.4 Graphene black hole**

where

$$\hat{k}\_x = -i\frac{\partial}{R\partial\theta}, \quad \hat{k}\_\gamma = -i\frac{\partial}{\partial\mathbf{y}}, \quad \hat{\sigma}\_x \left(\overrightarrow{r}\right) = \hat{\sigma}\_x \cos\theta - \hat{\sigma}\_x \sin\theta. \tag{22}$$

Next one can take

$$\gamma = -\frac{\sqrt{3}}{2} a V\_{pp}^{\pi}, \quad \gamma' = -\frac{\sqrt{3}}{2} a \left( V\_{pp}^{\sigma} - V\_{pp}^{\pi} \right), \quad p = 1 - \frac{3\gamma'}{8\chi}, \tag{23}$$

where *a* is the length of the atomic bond and *V<sup>σ</sup> pp, V<sup>π</sup> pp* are the hopping integrals for the *σ* and *π* bond, respectively.

For the interatomic source of the SOC, one has

$$\delta = \frac{\Delta}{3\varepsilon\_{\pi\sigma}}, \quad \Delta = i \frac{\Im \hbar}{4m^2 c^2} \left\langle \varkappa\_l | \frac{\partial V}{\partial \varkappa} \hat{p}\_j - \frac{\partial V}{\partial \jmath} \hat{p}\_x | y\_l \right\rangle \tag{24}$$

with the difference of energies of the relevant *π* and *σ* orbitals

$$
\epsilon\_{\pi\sigma} = \epsilon\_{2p}^{\pi} - \epsilon\_{2p}^{\sigma} \tag{25}
$$

*xl*, and *yl* being the local coordinates. By applying the transformation

$$
\hat{H}' = \hat{U}\hat{H}\hat{U}^{-1}, \qquad \hat{U} = \begin{pmatrix}
\exp\left(i\hat{\sigma}\_{\mathcal{V}}\frac{\theta}{2}\right) & 0 \\
& \mathbf{0} & \exp\left(i\hat{\sigma}\_{\mathcal{V}}\frac{\theta}{2}\right) \\
\mathbf{0} & \exp\left(i\hat{\sigma}\_{\mathcal{V}}\frac{\theta}{2}\right)
\end{pmatrix} \tag{26}
$$

the transformed Hamiltonian *H*^ <sup>0</sup> will have the form with two terms, including the *H*^ *SOC* term which corresponds to the spin-orbit coupling

$$\hat{H}' = \hat{H}\_{\text{kin}} + \hat{H}\_{\text{SOC}} \hat{H}\_{\text{kin}} = -i\gamma \left( \partial\_{\text{\mathcal{I}}} \text{Id}\_{2} \otimes \hat{\imath}\_{\text{\mathcal{I}}} + \frac{1}{R} \partial\_{\text{\mathcal{I}}} \text{Id}\_{2} \otimes \hat{\imath}\_{\text{\mathcal{I}}} \right) \hat{H}\_{\text{SOC}} = \lambda\_{\text{\mathcal{I}}} \hat{\alpha}\_{\text{x}} \otimes \hat{\imath}\_{\text{\mathcal{I}}} - \lambda\_{\text{x}} \hat{\alpha}\_{\text{\mathcal{I}}} \otimes \hat{\imath}\_{\text{\mathcal{I}}} \tag{27}$$

The operators ^*sx, y, <sup>z</sup>* are the Pauli matrices, which transform the wave function of the *A* sublattice into the wave function of the *B* sublattice and vice versa.

In our model, the SOC is induced by the curvature, and it is described with the help of two strength parameters, namely, *λ<sup>x</sup>* and *λ<sup>y</sup>* in the form

$$
\lambda\_{\mathbf{x}} = \frac{\gamma}{R} \left(\frac{1}{2} + 2\delta p\right), \quad \lambda\_{\mathbf{y}} = -\frac{\delta \mathbf{y}'}{4R}, \tag{28}
$$

for the case of single-wall carbon nanotube with different magnitude. Here, ∣*λy*∣≪∣*λx*∣ and for *R* ! 0, both strengths go to infinity. So reminding the previous results, the chiral massive fermions should be detected around the wormhole bridge. For more complicated forms as perturbed nanotube in the wormhole center, the geometry of the corresponding graphene sheets will be curved, and this brings a significant change of the physical properties.
