**1.4 Graphene black hole**

The expression ^*f* has the following form

*R∂θ*

ffiffiffi 3 p <sup>2</sup> *aV<sup>π</sup>*

for the *σ* and *π* bond, respectively.

where

^ *kx* ¼ �*<sup>i</sup> <sup>∂</sup>*

Next one can take

*γ* ¼ �

^*<sup>f</sup>* <sup>¼</sup> *<sup>γ</sup>* ^

*,* ^

*kx* � *<sup>i</sup>*^ *ky* � �

> *∂ ∂y*

> > ffiffiffi 3 p <sup>2</sup> *a V<sup>σ</sup>*

*,* <sup>Δ</sup> <sup>¼</sup> *<sup>i</sup>* <sup>3</sup><sup>ℏ</sup>

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

*, <sup>U</sup>*^ <sup>¼</sup>

the *H*^ *SOC* term which corresponds to the spin-orbit coupling

<sup>ϵ</sup>*πσ* <sup>¼</sup> <sup>ϵ</sup>*<sup>π</sup>*

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

0

BBB@

<sup>4</sup>*m*<sup>2</sup>*c*<sup>2</sup> *xl*<sup>j</sup>

<sup>2</sup>*<sup>p</sup>* � <sup>ϵ</sup>*<sup>σ</sup>*

exp *iσ*^*<sup>y</sup>*

1

� �

The operators ^*sx, y, <sup>z</sup>* are the Pauli matrices, which transform the wave function of

In our model, the SOC is induced by the curvature, and it is described with the

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 *A* sublattice into the wave function of the *B* sublattice and vice versa.

help of two strength parameters, namely, *λ<sup>x</sup>* and *λ<sup>y</sup>* in the form

*<sup>λ</sup><sup>x</sup>* <sup>¼</sup> *<sup>γ</sup> R* 1 <sup>2</sup> <sup>þ</sup> <sup>2</sup>*δ<sup>p</sup>* � �

*<sup>R</sup> <sup>∂</sup>θ*Id2⊗^*sx*

*, <sup>λ</sup><sup>y</sup>* ¼ � *δγ*<sup>0</sup>

*θ* 2 � �

0 exp *iσ*^*<sup>y</sup>*

*ky* ¼ �*i*

*pp, γ*<sup>0</sup> ¼ �

where *a* is the length of the atomic bond and *V<sup>σ</sup>*

For the interatomic source of the SOC, one has

*<sup>δ</sup>* <sup>¼</sup> <sup>Δ</sup> 3*επσ*

*<sup>H</sup>*^ <sup>0</sup> <sup>¼</sup> *<sup>U</sup>*^ *<sup>H</sup>*^ *<sup>U</sup>*^ �<sup>1</sup>

the transformed Hamiltonian *H*^ <sup>0</sup>

**100**

*<sup>H</sup>*^ <sup>0</sup> <sup>¼</sup> *<sup>H</sup>*^ *kin* <sup>þ</sup> *<sup>H</sup>*^ *SOC, <sup>H</sup>*^ *kin* ¼ �*i<sup>γ</sup> <sup>∂</sup>y*Id2⊗^*sy* <sup>þ</sup>

þ *i δγ*0 <sup>4</sup>*<sup>R</sup> <sup>σ</sup>*^*<sup>x</sup> <sup>r</sup>* !� �

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

*, σ*^*<sup>x</sup> r* !� �

> *pp* � *<sup>V</sup><sup>π</sup> pp* � �

> > *∂V <sup>∂</sup><sup>x</sup> <sup>p</sup>*^*<sup>y</sup>* � *<sup>∂</sup><sup>V</sup>*

� <sup>2</sup>*δγ<sup>p</sup>*

*<sup>R</sup> <sup>σ</sup>*^*y,* (21)

<sup>8</sup>*<sup>γ</sup> ,* (23)

(24)

(26)

(27)

*pp* are the hopping integrals

¼ *σ*^*<sup>x</sup>* cos *θ* � *σ*^*<sup>z</sup>* sin *θ:* (22)

*, p* <sup>¼</sup> <sup>1</sup> � <sup>3</sup>*γ*<sup>0</sup>

*<sup>∂</sup><sup>y</sup> <sup>p</sup>*^*x*j*yl*

<sup>2</sup>*p,* (25)

*θ* 2 � � 1

CCCA

*, <sup>H</sup>*^ *SOC* <sup>¼</sup> *<sup>λ</sup>yσ*^*x*⊗^*sy* � *<sup>λ</sup>xσ*^*y*⊗^*sx:*

<sup>4</sup>*<sup>R</sup> ,* (28)

0

will have the form with two terms, including

*pp, V<sup>π</sup>*

� �

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 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 the defects with the applications in cosmological models.
