# Strain effects on the electronic properties of a graphene wormhole

J. E. G. Silva<EMAIL_ADDRESS>Universidade Federal do Ceará,
Departamento de Física, 60455-760, Fortaleza, CE, Brazil  Ö. Yeşiltaş
<EMAIL_ADDRESS>Department of Physics, Faculty of Science, Gazi
University, 06500 Ankara, Turkey  J. Furtado<EMAIL_ADDRESS>Universidade Federal do Cariri, Centro de Ciências e Tecnologia, 63048-080,
Juazeiro do Norte, CE, Brazil  Department of Physics, Faculty of Science, Gazi
University, 06500 Ankara, Turkey  A. A. Araújo Filho<EMAIL_ADDRESS>Departamento de Física Teórica and IFIC, Centro Mixto Universidad de
Valencia–CSIC. Universidad de Valencia, Burjassot-46100, Valencia, Spain
Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008,
58051-970, João Pessoa, Paraíba, Brazil

(August 28, 2024)

###### Abstract

In this work, we explore the strain and curvature effects on the electronic
properties of a curved graphene structure, called the graphene wormhole. The
electron dynamics is described by a massless Dirac fermion containing
position–dependent Fermi velocity. In addition, the strain produces a
pseudo–magnetic vector potential to the geometric coupling. For an isotropic
strain tensor, the decoupled components of the spinor field exhibit a
supersymmetric (SUSY) potential, depending on the centrifugal term and the
external magnetic field only. In the absence of a external magnetic field, the
strain yields to an exponential damped amplitude, whereas the curvature leads
to a power–law damping of the wave function. The spin–curvature coupling
breaks the chiral symmetry between the upper and the lower spinor component,
which leads to the increasing of the wave function on either upper or lower
region of the wormhole, i.e., depending on the spin number. By adding an
uniform magnetic field, the effective potential exhibits an asymptotic
quadratic profile and a spin–curvature barrier near the throat. As a result,
the bound states (Landau levels) are confined around the wormhole throat
showing an asymmetric and spin–dependent profile.

##  I Introduction

Two dimensional materials, such as graphene geim , silicene silicene  and
phosphorene phosphorene , have been the subject of intense investigations due
to their outstanding properties. Beyond the remarkable mechanical katsnelson
and electronic properties Novoselov2004 ; electronic , graphene can also be
seen as a table–top laboratory for relativistic physics. Indeed, since the
conduction electrons are effectively described as massless Dirac fermions,
relativistic effects such as zitterbewegung zitter , Klein tunneling klein
and atomic collapse collapse  have been observed. Since the graphene layer can
assume a curved shape, the curvature effects might lead to new interesting
relativistic effects, such as the Hawking–Unruh effect hawking ; hawking2 .

The study of a Dirac fermion confined into a two dimensional surface was
initially addressed in Ref.BJ  and further developments were provide
afterwards diracsurface ; diracsurface2 ; diracsphere . For a relativistic
fermion intrinsically living on a curved surface, a physical realization was
found for conducting electrons on two dimensional carbon–based structures, as
the fullerenes diracintrinsic ; diracintrinsic1 , carbon nanotubes saito  and
graphitic cones cone . In graphene, the massless Dirac equation in curved
spaces was studied in a variety of shapes, such as the localized gaussian bump
contijo , the cone furtado , a helical graphene ribbon atanasov ; watanabe , a
corrugated plane corrugated , a Möbius ring mobius1 ; mobius2 , a torus ozlem
; Yesiltas:2021crm  among others. The surface curvature produces a
spin–curvature coupling which leads to a geometric Aharonov–Bohm–like effect
geometricphase , a modified spin–orbit coupling geometricsoc ; geometricsoc2
and a geometric spin–Hall effect geometricmonopole .

Besides the curvature, the deformations of the graphene layer modify the
effective Dirac fermion dynamics as well, producing the so–called
pseudo–magnetic fields ribbons . This vector potential steams from the strain
tensor defined by the deformations of the graphene layer and the
pseudo–magnetic term comes from the coupling to the Dirac fermion; it is
similar to the minimal coupling to a magnetic field gaugestrain . The strain
applied to graphene can mimic a strong magnetic field strainstrong  and lead
to important applications strainappli . From the strain tensor, an effective
Hamiltonian for the Dirac fermion was derived using the tight–binding approach
vozmediano , containing an anisotropic and position–dependent Fermi velocity
and a pseudo–magnetic strain vector in the continuum limit. Since the strain
may contain both in–plane and out–of–plane components, the effective Dirac
Hamiltonian was extended in order to encompass all the stretching and bending
effects vozmediano2 . In addition, the quantum field interaction of the
effective Dirac fermion and the strain was discussed in sinner ;
gaugegrapheneqft .

An interesting curved graphene structure is the so–called graphene wormhole,
where two flat graphene layers are connected by a carbon nanotube wormhole .
Since its shape (cylinder) has a non–vanishing mean curvature, the
discontinuity of the curvature at the graphene–nanotube junction leads to
modifications of the energy spectrum and the possibility of localized states
near to it graphenejunction . Although, the Dirac fermions on the upper and
lower layers are free states (non–normalizable), the curvature of the nanotube
allows the existence of normalizable zero–modes confined at the radius of the
wormhole picak ; wormhole3 . In order to avoid the discontinuity at the
junction, a smooth graphene wormhole was proposed considering a continuum and
asymptotic flat catenoid surface dandoloff ; euclides ; deSouza:2022ioq . The
negative curvature of the catenoid leads to a repulsive spin–curvature
coupling near the wormhole throat, allowing only the zero–mode as a localized
state around the throat wormhole4 ; ozlem2 ; wormhole5 .

In this work, we consider the effects of the curvature and the strain on the
effective Dirac fermion living in a catenoid–shaped graphene wormhole. We
extend the effective Hamiltonian obtained in vozmediano  to a curved surface,
introducing the usual spin–connection coupling. We explore the different
effects driven by the curvature, isotropic strain and an external magnetic
field. Since the surface is asymptotic flat, the lattice deformation which
yields the curvature and strain should be concentrated around the throat. The
strain leads to a vector potential along the surface meridian, whereas the
spin–curvature coupling points in the angular direction. Moreover, the strain
vector potential provides an exponential damping of the wave function, whereas
the curvature leads to a power–law decay. By adopting the so-called
supersymmetric quantum mechanic-like approach ozlem2 , the spinor components
exhibit a chiral symmetry breaking. Indeed, the upper component has its
probability density enhanced near the wormhole throat in the upper layer,
whereas the lower component is enhanced in the lower layer. The ground state
zero mode also exhibits this chiral behaviour, since it is exponentially
damped either in the upper or in the lower layer depending on the total
angular momentum. By applying an uniform magnetic field, the Landau levels are
also modified by the curved geometry and strain, leading to asymmetric
localized states near the throat.

This work is organized as the follows. In the section (II), we provide a brief
review on the catenoid-shape graphene wormhole geometry. In the section (III)
we present the effective Hamiltonian containing the strain, curvature and
external magnetic field interactions. The section (IV) is devoted to the
symmetries of the effective Hamiltonian and in the section (V) we employ the
SUSY-QM approach in order to investigate the effects of each interaction.
Finally, additional discussion and perspectives are outlined in the section
(VI).

##  II Graphene wormhole geometry

In this section, we define the graphene wormhole surface and describe some of
its most important properties. We consider a smooth surface connecting an
upper to the lower layer (flat planes). For this purpose, we choose a catenoid
shaped surface. In other words, the catenoid surface can be describe in
coordinates by euclides ; ozlem2

$\vec{r}(u,\phi)=\sqrt{R^{2}+u^{2}}\left(\cos\phi\hat{i}+\sin\phi\hat{j}\right)+R\sinh^{-1}\left(\frac{u}{R}\right)\hat{k},$
(1)

where $R$ is the throat radius, $-\infty<u<\infty$ describes the meridian
coordinate and $\varphi$ is the parallel coordinate $\varphi\in[0,2\pi)$, as
shown in Fig.1.

Figure 1: Graphene wormhole geometry. The meridian coordinate $u$ connects the
lower to the upper asymptotic flat regions.

The tangent vectors are given by

$\displaystyle\vec{e}_{1}$ $\displaystyle=$
$\displaystyle\frac{\partial\vec{r}}{\partial
u}=\frac{1}{\sqrt{u^{2}+R^{2}}}(u\hat{r}+R\hat{k})$ (2)
$\displaystyle\vec{e}_{2}$ $\displaystyle=$
$\displaystyle\frac{\partial\vec{r}}{\partial\phi}=\sqrt{u^{2}+R^{2}}\hat{e}_{2},$
(3)

where $\hat{e}_{2}=\cos\phi\hat{i}+\sin\phi\hat{j}$ is the unit vector along
the $\phi$ direction. From the tangent vectors $(\hat{e}_{1},\hat{e}_{2})$, we
can define the surface induced metric $g_{ij}=\vec{e}_{i}\cdot\vec{e}_{j}$. In
$(u,\phi)$ coordinates, the surface metric takes the form
$g_{ij}=diag(1,(R^{2}+u^{2}))$. Thus, the $2+1$ spacetime interval has the
form euclides

$\mathrm{d}s^{2}=\mathrm{d}t^{2}-\mathrm{d}u^{2}-(R^{2}+u^{2})\mathrm{d}\phi^{2},$
(4)

where we adopt the $(+,-,-)$ spacetime metric signature convention. Note that
the line element in Eq.(4) is invariant under time–translations and rotations
with respect to the $\mathrm{z}$ axis (axissymmetric).

Let us now obtain the the main geometric quantities for the electron dynamics,
namely, the dreinbeins, connections and curvature. The dreinbeins are related
to the spacetime metric by the relation

$g_{\mu\nu}=e^{a}_{\mu}e^{b}_{\nu}\eta_{ab}.$ (5)

Thus, for the catenoid, the only non–vanishing components of the dreinbeins
are

$\displaystyle e^{a}_{\mu}$ $\displaystyle=$ $\displaystyle
diag(1,1,\sqrt{R^{2}+u^{2}})$ (6)

Remember that they modify the Fermi velocity by turning it into a position
dependet configuration. Moreover, from the dreinbeins, we can defined the
moving frame $\theta^{a}=e^{a}_{\mu}\mathrm{d}x^{\mu}$, where, in the
catenoid, take the form $\theta^{0}=\mathrm{d}t$, $\theta^{1}=\mathrm{d}u$,
$\theta^{2}=\sqrt{R^{2}+u^{2}}\mathrm{d}\phi$. From the torsion–free
condition, i.e.,
$T^{a}=\mathrm{d}\theta^{a}+\omega^{a}_{b}\wedge\theta^{b}=0$, the only
non–vanishing one–form connection coefficient
$\omega^{a}_{b}=\Gamma_{cb}^{a}\theta^{c}$ is given by

$\displaystyle\omega^{2}_{1}$ $\displaystyle=$
$\displaystyle\frac{u}{R^{2}+u^{2}}\theta^{2}.$ (7)

The curvature 2–form
$R^{a}_{a}=d\omega^{a}_{b}+\omega^{a}_{c}\wedge\omega^{c}_{b}$ has only
non–vanishing component, namely
$R^{2}_{1}=-\frac{R^{2}}{(R^{2}+u^{2})^{2}}\theta^{2}\wedge\theta^{1}$.
Accordingly, the gaussian curvature $K=\delta^{ab}R_{ab}$ has the form

$\displaystyle K=-\frac{R^{2}}{(R^{2}+u^{2})^{2}}.$ (8)

Here, it is important to point out that the catenoid has a negative Gaussian
curvature concentrated around the throat and it vanishes in the regions far
from it. In Fig. 2, we display the behavior of graphene wormhole curvature
$K$. Note that the surface is asymptotically flat. Thus, the effects of the
curved geometry and strain on the electron should be concentrated around the
origin. Furthermore, as $R\rightarrow 0$, the curvature tends to a $\delta(r)$
function, as it is reported in the literature for a discontinuous graphene
wormhole wormhole ; picak ; wormhole3 .

Figure 2: The Gaussian curvature $K(u)$ of the graphene wormhole. The
curvature is smooth and concentrated around the throat of the wormhole. For
$R\rightarrow 0$, the curvature tends to a delta–like function.

##  III Strain Hamiltonian

After a brief review of the main geometric properties of the graphene
wormhole, let us now discuss the effective Hamiltonian containing the strain
and curvature effects on the electron. We follow closely the Ref.(vozmediano )
where the most general effective Hamiltonian was derived. The effective
Hamiltonian for a Dirac fermion constrained to a flat surface under influence
of the strain and an external magnetic field was found in Ref. (vozmediano ).
We propose a generalization of the effective Hamiltonian of Ref.(vozmediano )
in the continuum limit for curved surfaces in the form

$\mathcal{H}_{D}=-i\hbar\left(v_{i}^{j}\sigma^{i}\partial_{j}+ie\sigma^{i}A_{i}+v_{0}\sigma^{i}\Gamma_{i}+v_{0}\sigma^{i}\Omega_{i}\right),$
(9)

where $v_{i}^{j}$ is a position–dependent Fermi velocity tensor defined in
terms of the strain tensor $u_{ij}$ as vozmediano

$v_{i}^{j}=v_{0}\left[\delta_{i}^{j}-\frac{\beta}{4}(2u_{i}^{j}+\delta_{i}^{j}u_{k}^{k})\right],$
(10)

and $v_{0}=\frac{3t_{0}a}{2}$ is the undeformed Fermi velocity, $t$ is the
hopping parameter, $a$ is the lattice constant and $\beta=|\partial\ln
t/\partial\ln a|$ gaugestrain . The definition of the strain tensor will be
given in the next subsection. Note that, when $\beta=0$, the usual constant
Fermi velocity is recovered. Besides, the tensor nature of $v_{ij}$ means that
the Fermi velocity depends on the direction on the surface.

In addition, the strain on the surface also induces a new vector field, called
the strain vector $\Gamma_{i}$, as a divergence of the velocity tensor
vozmediano . Thus, for a curved surface, it is defined as

$\Gamma_{i}=\frac{1}{2v_{0}}\nabla_{j}v^{j}_{i},$ (11)

where the definition of the strain vector in Eq.(11) is independent of the
coordinate choice.

The curved Pauli matrices are defined as watanabe ; mobius1 ; mobius2

$\sigma^{i}=e^{i}_{a}\sigma^{a},$ (12)

where $\sigma^{a}$ are the usual flat Pauli matrices, and $e^{i}_{a}$ are the
zweinbeins matrices which satisfy

$g_{ij}=e^{a}_{i}e^{b}_{j}\delta_{ab}.$ (13)

The definition of the curved sigma matrices employed in Eq.(12) ensures that
these matrices do not depend on the particular choice of coordinates of the
surface (surface covariance). It is worthwhile to mention that the definitions
of the velocity tensor in Eq.(24 and the curved Pauli matrices in Eq.(12) lead
to a position and direction dependent Dirac kinetic term
$H_{1}=v_{i}^{j}\sigma^{i}\partial_{j}$.

Figure 3: Stress function $\sigma(u)$ for

Figure 4: Fermi velocity $v(u)$ for $\beta=0.1$

In the effective Hamiltonian exhibited in Eq.(9), $A_{i}$ is the external
magnetic potential and $\Omega_{i}$ is the spinor connection furtado ; mobius2
; ozlem2

$\Omega_{i}=\frac{1}{4}\omega_{i}^{ab}\gamma_{a}\gamma_{b}.$ (14)

The curved $\gamma^{\mu}$ matrices are related to the flat $\gamma^{a}$ ones
by the dreinbeins $e^{a}_{\mu}$, i.e., $\gamma^{\mu}=e^{\mu}_{a}\gamma^{a}$.
The dreinbeins are defined as $g_{\mu\nu}=e_{\mu}^{a}e_{\nu}^{b}$. In $(2+1)$
dimension, we can adopt the following representation to the flat Dirac
$\gamma^{a}$ matrices $\gamma_{0}=\sigma_{3}$, $\gamma_{1}=-i\sigma_{2}$ and
$\gamma_{2}=-i\sigma_{1}$ mobius2 ; ozlem ; ozlem2 . Thus, the curved Dirac
matrices on the wormhole graphene surface have the form

$\displaystyle\gamma^{t}$ $\displaystyle=$ $\displaystyle
e^{t}_{0}\gamma^{0}=\gamma_{0},$ (15) $\displaystyle\gamma^{u}$
$\displaystyle=$ $\displaystyle\gamma^{1},$ (16) $\displaystyle\gamma^{\phi}$
$\displaystyle=$ $\displaystyle
e_{2}^{\phi}\gamma^{2}=\frac{1}{\sqrt{R^{2}+u^{2}}}\gamma^{2}.$ (17)

From the connection 1–form in Eq.(7), only $\omega^{2}_{1}$ is non zero. Thus,
the only non–vanishing component of the spinor connection $\Omega_{\mu}$ is

$\Omega_{\varphi}=\frac{i}{2}\frac{u}{\sqrt{R^{2}+u^{2}}}\sigma_{3}.$ (18)

Note that, since $-\infty<u<\infty$, the geometric spinor potential in Eq.(18)
is an odd function under parity. This parity violation does not occur for the
Dirac fermion in a flat plane diracplanar  or the graphitic cone furtado .
Moreover, for $R=0$ or for $R\neq 0$ and $u\rightarrow\pm\infty$, the
geometric connection is constant, as found for conic surfaces furtado . It is
worth mentioning that, due to the resemblance of the spinor and gauge field
coupling, the spinor potential is sometimes interpreted as a kind of a
pseudo–magnetic potential steaming from the curved geometry contijo .

Furthermore, the strain produces two different potentials on the Dirac
electron. The first potential, steaming from the strain in Eq.(11), is a
vector potential, whereas the second one in Eq.(18) is a spinorial potential
depending on the surface connection. In the next subsections, we choose a
particular configuration for the strain and the external magnetic field and
explore the differences between these three interactions.

###  III.1 Strain tensor

Now, let us investigate how the strain tensor $u_{ij}$ modifies the catenoid
surface. In order to do it, we assume that the tensions over the surface are
static and isotropic. Therefore, we consider the non–uniform isotropic stress
tensor in the form

$\sigma_{j}^{i}=\sigma(u)\delta_{j}^{i}.$ (19)

Since the catenoid brige is an asymptotically flat surface, we are interested
in stress tensor which vanishes away from the throat and it is finite at the
origin, i.e.,

$\displaystyle\lim_{u\rightarrow 0}\sigma(u)$ $\displaystyle=$
$\displaystyle\sigma_{0},$ (20)
$\displaystyle\lim_{u\rightarrow\pm\infty}\sigma(u)$ $\displaystyle=$
$\displaystyle 0,$ (21)

where $\sigma_{0}$ is a constant, which accounts for the maximum value of the
surface tension. These conditions guarantee a stress tensor concentrated
around the catenoid throat. Indeed, since the surface is asymptotic flat, the
curvature and strain effects should vanish as $u\rightarrow\infty$.

Figure 5: Geometric connection $\Omega_{\phi}$ for $R=0.1$ and $R=1$.

Figure 6: Strain vector $\Gamma_{u}$ for $R=0.1$ and $R=1$.

We assume that the mechanical properties of the surface are in the linear
elastic regime. Thereby, the stress and the strain tensors are related by
vozmediano

$\sigma_{ij}=\lambda\theta g_{ij}+2\mu u_{ij},$ (22)

with $\lambda$ and $\mu$ are the Lamé coefficients and $\theta=u^{i}_{i}$ is
the trace of the strain tensor. From the ansatz employed in Eq.(19), we obtain
the strain tensor as

$u^{i}_{j}=\frac{1}{2(\lambda+\mu)}\sigma(u)\delta^{i}_{j}.$ (23)

The form of the strain tensor in Eq.(23) shows that the deformations undergone
by the surface are isotropic and concentrated aroun the catenoid throat. The
position–dependent Fermi velocity tensor $v_{ij}$ can be written as

$v_{i}^{j}=v(u)\delta_{i}^{j},$ (24)

where the position–dependent Fermi velocity function $v(u)$ is given by

$v(u)=v_{0}\left[1-\frac{\beta}{2}\frac{\sigma(u)}{\lambda+\mu}\right].$ (25)

Accordingly, the components of the pseudo–vector potential $\Gamma_{i}$ are

$\displaystyle\Gamma_{u}$ $\displaystyle=$
$\displaystyle-\frac{\beta}{4}\frac{\sigma^{\prime}(u)}{\lambda+\mu},$ (26)
$\displaystyle\Gamma_{\phi}$ $\displaystyle=$ $\displaystyle 0.$ (27)

In this work, we assume the isotropic stress function $\sigma(u)$ as

$\sigma(u)=\sigma_{0}\frac{R^{2}}{R^{2}+u^{2}}.$ (28)

We see that $\sigma(u)$ becomes even more concentrated when $R\rightarrow 0$,
as it is shown in Fig. (4). On the other hand, in Fig. (4), we show the plot
of the Fermi velocity function $v(u)$ for different values of $\beta$.
Remarkably, it decreases for the regions close to the wormhole throat (high
curvature). Such a feature was already found in a smooth ripple curved
graphene layer contijo .

In addition, the behavior of the geometric spinor connection
$\Omega_{\varphi}$ and the strain vector $\Gamma_{u}$ are shown in Fig.(6) and
Fig.(6), respectively. Note that both terms are parity odd functions with
respect to the $u$ coordinate. In this sense, both potentials yield to barrier
between the lower $u<0$ and upper $u>0$ layers. However, despite this
similarity, the strain potential given by Eq.(11) and the spinor potential
given by Eq.(18) have different natures, forms and components. Therefore,
these potentials produce different effects on the Dirac electron, as we shall
see in the next section.

###  III.2 Magnetic vector potential

Now, let us see how the external magnetic field $\vec{B}$ modifies the
Hamiltonian. For an uniform magnetic field $\vec{B}=B\hat{k}$, the vector
potential $\vec{A}$ is given by $\vec{A}=\frac{1}{2}\vec{B}\times\vec{r}$.
Using the coordinates system in Eq.(1), we obtain

$\vec{A}=\frac{B}{2}\sqrt{R^{2}+u^{2}}\hat{e}_{2}.$ (29)

For $R=0$, the expression for the vector potential in Eq.(29) reduces to
$\vec{A}=\pm\frac{B}{2}u\hat{e}_{2}$, as found in Ref.(diracmagnetic1 ). For
$u\gg R$, it yields to $A^{2}\approx\frac{B}{2}u$, as in a conical surface
diracmagnetic1  and on the flat plane diracplanar . In addition, at $u=0$, it
has a finite value $A^{2}=\frac{BR}{2}$. Since
$\vec{e}_{\phi}=\sqrt{R^{2}+u^{2}}\hat{e}_{2}$, the component of $\vec{A}$ in
coordinates is given by $A^{\phi}=\frac{B}{2}$. Accordingly, the contravariant
component $A_{\phi}=g_{\phi\phi}A^{\phi}$, we have

$A_{\phi}=\frac{B}{2}(R^{2}+u^{2}).$ (30)

A similar expression for the vector potential was found for the discontinuous
graphene wormhole wormhole3 , except for the presence of the throat radius
$R$. The electromagnetic potential displayed in Fig.(10) is parity even, in
contrast with the geometric spinor connection shown in Fig.(8). Also, the
strain vector in Fig.(8) turns out to have a parity odd configuration.

Figure 7: Spin connection $\Omega_{\phi}$ for $R=0.1$ and $R=1$. This
curvature potential exhibit a parity–odd angular configuration.

Figure 8: Strain vector $\Gamma_{u}$ for $R=0.1$ and $R=1$. The strain–driven
potential has a components only along the meridian.

##  IV Effective Hamiltonian

Once we have discussed all those interactions acting upon the electron, i.e.,
the curved geometry, strain and external magnetic field, let us now obtain the
respective Hamiltonian. By collecting all the interactions, the effective
Hamiltonian becomes

$\displaystyle\mathcal{H}_{D}$ $\displaystyle=$ $\displaystyle-i\hbar
v_{0}\left\\{\sigma_{1}\left[v(u)\partial_{u}+\bar{\beta}\sigma^{\prime}+\frac{u}{2(R^{2}+u^{2})}\right]\right.$
(31)
$\displaystyle\left.-\frac{i\sigma_{2}}{\sqrt{R^{2}+u^{2}}}[v(u)\partial_{\phi}+eA_{\phi}]\right\\},$

where $\bar{\beta}=\frac{\beta}{4\lambda+\mu}$. Notice that the strain vector
and the spinor connection modifies the Dirac equation leading to a canonically
momentum on the $u$ of form

$\hat{P}_{u}=-i\hbar
v_{0}\left[v(u)\partial_{u}+\bar{\beta}\sigma^{\prime}+\frac{u}{2(R^{2}+u^{2})}\right].$
(32)

Additionally, along the angular $\phi$ direction, the canonically conjugate
momentum is modified by

$\hat{P}_{\phi}=-i\hbar
v_{0}\frac{1}{\sqrt{R^{2}+u^{2}}}[v(u)\partial_{\phi}].$ (33)

In this manner, the effective Hamiltonian can be rewritten in the familiar
form $\mathcal{H}_{D}=v_{0}\vec{\sigma}\cdot(\vec{P}-e\vec{A}),$ where
$\vec{P}=(P_{u},P_{\phi})$ are the canonically conjugate momenta and
$\vec{\sigma}=(\sigma_{1},\sigma_{2})$ are the flat Pauli matrices.

The expression in Eq.(31) depends only on the coordinate $u$. The symmetry of
the Hamiltonian with respect to the angular $\phi$ variable is the result of
the surface axial symmetry. Thus, the wave function should also inherit this
symmetry. In fact, consider the angular momentum operator with respect to the
$\mathrm{z}$ axis, $\hat{L}_{z}=-i\hbar\frac{\partial}{\partial\phi}$ such
that, $\hat{L}_{z}\psi=\hbar l\psi$, where $l$ is the orbital angular momentum
with respect to the $\mathrm{z}$ axis. For a non–relativistic and spinless
electron on the graphene wormhole, an axissymmetric wave function can be
written as $\psi(u,\phi)=e^{il\phi}\psi(u)$ euclides . However, as it is
well–known for the relativistic electron, $\hat{L}_{z}$ no longer commute with
$\mathcal{H}_{D}$, although the total angular momentum operator along the $z$
direction $\hat{J}_{z}=\hat{L}_{z}+\hat{S}_{z}$ does diracplanar . Since
$\hat{S}_{3}=\frac{\hbar}{4}[\gamma^{1},\gamma^{2}]$, then the spin operator
with respect to the $\mathrm{z}$ axis is given by

$S_{z}=-i\frac{\hbar}{2}\sigma_{3}.$ (34)

Here, the total angular momentum operator has the form
$\hat{J}_{z}=-i\hbar\left(\frac{\partial}{\partial\phi}+\frac{1}{2}\sigma_{3}\right)$,
where $\hat{J}\psi=m\hbar\psi$ and $m=l\pm 1/2$ diracplanar2 . Therefore,
considering the axial symmetry on the spinorial wave function, so that
wormhole4 ; diracplanar2

$\Psi(u,\phi)=e^{im\phi}\psi(u),$ (35)

the Dirac equation $\mathcal{H}_{D}\Psi=E\Psi$ leads to the Dirac equation
$\tilde{\mathcal{H}}_{D}\psi=E\psi$, in which the effective Hamiltonian
simplifies to

$\displaystyle\tilde{\mathcal{H}}_{D}$ $\displaystyle=$ $\displaystyle-i\hbar
v_{0}\left(\begin{array}[]{cc}0&v(u)\partial_{u}+\bar{\beta}\sigma^{\prime}+\frac{u}{2(R^{2}+u^{2})}-\frac{[v(u)m+eA_{\phi}]}{\sqrt{R^{2}+u^{2}}}\\\
v(u)\partial_{u}+\bar{\beta}\sigma^{\prime}+\frac{u}{2(R^{2}+u^{2})}+\frac{[v(u)m+eA_{\phi}]}{\sqrt{R^{2}+u^{2}}}&0\end{array}\right).$
(38)

Since the effective Hamiltonian in Eq.(38) has parity violating terms steaming
from the geometric connection and the strain vector, we can not assume that
the spinor $\psi=\left(\begin{array}[]{cc}\psi_{1}\\\
\psi_{2},\end{array}\right)$ is parity invariant. This parity violation of the
Dirac spinor on the graphene wormhole is in contrast with the relativistic
electron on a flat plane diracplanar ; diracplanar2 .

In Eq.(38), the position–dependent velocity function $v(u)$ multiplies the
partial derivative $\partial_{u}$. By writing
$v\frac{d}{du}=\frac{d}{d\zeta}$, for $v(u)$ given by Eq.(25), we have
$\zeta=u-\bar{\beta}R\,\tanh^{-1}\left(\frac{\sqrt{u}u}{\sqrt{\bar{\beta}-2}R}\right)$.
Unfortunately, this relation analytically can not be inverted, seeking to
rewrite Eq. (38) in terms of variable $\zeta$. Nevertheless, a graphic
analysis reveals only a small difference between $\zeta$ and $u$ as shown in
Fig. 12. Analogously, for the sake of simplicity, we adopt $\zeta\approx u$
from now on.

Figure 9: Vector potential angular component for an uniform magnetic field.

Figure 10: Magnetic vector potential $\vec{A}$ on the surface.

As a result, the Hamiltonian simplifies to

$\displaystyle\tilde{\mathcal{H}}_{D}$ $\displaystyle=$ $\displaystyle-i\hbar
v_{0}\left(\begin{array}[]{cc}0&\partial_{u}+\bar{\beta}\sigma^{\prime}+\frac{u}{2(R^{2}+u^{2})}-\frac{[m+eA_{\phi}]}{\sqrt{R^{2}+u^{2}}}\\\
\partial_{u}+\bar{\beta}\sigma^{\prime}+\frac{u}{2(R^{2}+u^{2})}+\frac{[m+eA_{\phi}]}{\sqrt{R^{2}+u^{2}}}&0\end{array}\right).$
(41)

The effective Hamiltonian in Eq.(41) shows clearly the distinctive interaction
terms steaming from the strain $\bar{\beta}\sigma^{\prime}$, from the
geometric connection $\frac{u}{2(R^{2}+u^{2})}$, centrifugal term
$\frac{m}{\sqrt{R^{2}+u^{2}}}$ and the electromagnetic coupling
$\frac{eA_{\phi}}{\sqrt{R^{2}+u^{2}}}$. In the next section, we explore the
effects of each interaction.

##  V Supersymmetric analysis

In this section, we employ a supersymmetric quantum mechanical approach ozlem
; ozlem2  to explore the features of the effective Hamiltonian in Eq.(41) and
find the solutions of the Dirac equation.

From the effective Eq.(41), it leads to

$\tilde{\mathcal{H}}_{D}\psi=\epsilon\psi,$ (42)

where the spinor $\psi=\left(\begin{array}[]{cc}\psi_{1}\\\
\psi_{2},\end{array}\right)$. The effective Dirac equation (42) can be written
as

$\displaystyle\left(\begin{array}[]{cc}0&i\mathcal{O}_{2}\\\
i\mathcal{O}_{1}&0\end{array}\right)\left(\begin{array}[]{cc}\psi_{1}\\\
\psi_{2},\end{array}\right)=\epsilon\left(\begin{array}[]{cc}\psi_{1}\\\
\psi_{2},\end{array}\right),$ (49)

where the firs–order operators $\mathcal{O}_{1,2}$ are defined as

$\mathcal{O}_{1,2}=\frac{\mathrm{d}}{\mathrm{d}u}+\bar{\beta}\sigma^{\prime}+\frac{u}{2(R^{2}+u^{2})}\pm\frac{(m+eA_{\phi})}{\sqrt{R^{2}+u^{2}}}.$
(50)

By performing the change on the wave function of the form

$\psi_{1,2}(u)=(R^{2}+u^{2})^{-1/4}e^{-\bar{\beta}\sigma(u)}\chi_{1,2}(u),$
(51)

the Dirac equation yields to a decoupled equations for the $\chi_{1}$ and
$\chi_{2}$ in a Klein–Gordon like form

$\displaystyle-\chi_{1,2}^{\prime\prime}+U_{eff1,2}^{2}\chi_{1,2}=\epsilon^{2}\chi_{1,2},$
(52)

where $\epsilon=\frac{E}{\hbar v_{0}}$ is the electron momentum and the
effective squared potential is given by

$U_{eff1,2}^{2}=\left(\frac{(m+eA_{\phi})}{\sqrt{R^{2}+u^{2}}}\right)^{2}\mp\left(\frac{(m+eA_{\phi})}{\sqrt{R^{2}+u^{2}}}\right)^{\prime}.$
(53)

The Klein–Gordon–like expression present in Eq. (52) has the structure of a
so–called supersymmetric quantum mechanics, whose superpotential $W$ is given
by

$\displaystyle W=\frac{m+eA_{\phi}}{\sqrt{R^{2}+u^{2}}}.$ (54)

Note that Eq.(54) is given by the spin–curvature potential
$\frac{m}{\sqrt{R^{2}+u^{2}}}$ and the magnetic coupling term
$\frac{eA_{\phi}}{\sqrt{R^{2}+u^{2}}}$ present in the Dirac equation.

Figure 11: Change of coordinate $\zeta=\zeta(u)$ for $\beta=1$.

Figure 12: Superpotential $W(u)$ for $R=1$, $B=0$.

The SUSY–like form of the effective squared potential

$U_{eff1,2}^{2}=W^{2}\mp W^{\prime}$ (55)

enables us to rewrite the decoupled system of second–order Klein–Gordon like
Eq.(52) as

$\displaystyle a^{\dagger}a\chi_{1}=\epsilon^{2}\chi_{1}$ (56) $\displaystyle
aa^{\dagger}\chi_{2}=\epsilon^{2}\chi_{2},$ (57)

where $a=\frac{d}{du}+W$ and $a^{\dagger}=-\frac{d}{du}+W$ are first–order
differential operators ozlem2 . The so-called superpartner squared-
Hamiltonians $H_{1}^{2}=a^{\dagger}a$ and $H_{2}^{2}=aa^{\dagger}$ satisfy
$H_{2}^{2}=(H_{1}^{2})^{\dagger}$ ozlem2 .

A remarkable feature of a quantum mechanical SUSY–like eq.(56) is the
existence of a nonvanishing ground state for $\epsilon=0$, known as the zero
mode wormhole ; picak . Indeed, for $\epsilon=0$, the conditions $a\chi_{1}=0$
and $a^{\dagger}\chi_{2}$ yield to

$\psi^{0}_{1,2}(u)=(R^{2}+u^{2})^{-1/4}e^{-\bar{\beta}\sigma(u)}e^{\mp\int{W(u^{\prime})du^{\prime}}}.$
(58)

Since the superpotential $W$ is related to the spin–curvature coupling and the
external magnetic field, the zero mode is related to the flux of curvature and
magnetic field near the throat wormhole .

The factor $(R^{2}+u^{2})^{-1/4}$ steams from the normalization condition on
the surface. Indeed, the normalization constant takes the form

$1=\int_{-b}^{b}\int_{0}^{2\pi}{||\psi||^{2}(R^{2}+u^{2})^{1/2}\mathrm{d}u\mathrm{d}\phi},$
(59)

where $-b<u<b$. Despite both the curved geometry and the strain reduce the
wave function amplitude for $u\rightarrow\pm\infty$, the curvature damps the
amplitude by a potential factor whereas the strain damps it by an exponential
factor.

Another noteworthy feature of the Dirac equation in curved surface is related
to the geometric phase furtado . Indeed, the holonomy operator
$U(\phi)=e^{\int_{0}^{\phi}{\Omega_{i}dx^{i}}}$, where $\Omega_{i}$ is the
spin connection in Eq.(18) leads to

$U(\phi)=e^{-\frac{i}{2}\frac{u}{R^{2}+u^{2}}\sigma_{3}\phi}.$ (60)

This geometric phase reflects the change on the wave function when the fermion
performs a $2\pi$ rotation for a given $u$ mobius2 . It is a kind of geometric
Aharonov–Bohm effect driven by the curvature instead of the magnetic field
furtado . By applying the geometric phase operator $U(\phi)$, i.e.,
$\psi^{\prime}=U(\phi)\psi$, the Dirac equation
$\tilde{\mathcal{H}}_{D}\psi=E\psi$ simplifies to
$\hat{\mathcal{H}}_{D}\psi^{\prime}=E\psi^{\prime}$, where the simplified
Hamiltonian $\hat{\mathcal{H}}_{D}$ is given by

$\displaystyle\tilde{\mathcal{H}}_{D}$ $\displaystyle=$ $\displaystyle-i\hbar
v_{0}\left(\begin{array}[]{cc}0&\partial_{u}+\bar{\beta}\sigma^{\prime}-\frac{[m+eA_{\phi}]}{\sqrt{R^{2}+u^{2}}}\\\
\partial_{u}+\bar{\beta}\sigma^{\prime}+\frac{[m+eA_{\phi}]}{\sqrt{R^{2}+u^{2}}}&0\end{array}\right).$
(63)

Thus, the curvature effects of the curved surface can be encoded into the
geometric phase in the operator $U(\phi)$ given by Eq.(60).

It is important to highlight that the effects of the strain, curved geometry
and the external magnetic fields are rather distinct. The strain and curved
geometry provide the geometric phase, whereas the centrifugal and the magnetic
field yield to the SUSY–like symmetry. In the following, we explore the
effects of the each term on the electronic states.

Figure 13: Effective squared potential for $m=1/2$ (on the left panel),
$m=-1/2$ (on the right panel) for $B=0$.

###  V.1 No external magnetic field

In the absence of a magnetic field, i.e., for $B=0$, the superpotential has a
simple form

$W(u)=\frac{m}{\sqrt{R^{2}+u^{2}}},$ (64)

whose behavior is plotted in Fig.12. The symmetric centrifugal barrier of the
superpotential yields to an asymmetric potential for $U^{2}_{eff}$, as shown
in Fig. 13. Note the dependence of the squared potential on the total angular
momentum $m$. This one is similar to that one encountered in the context of a
Dirac electron constrained to a helicoid potential watanabe .

Figure 14: Density of states for $\epsilon=1$ and $m=1/2$.

Figure 15: Density of states for $\epsilon=1$ and $m=-1/2$.

For $m\neq 0$, the Klein–Gordon like eq.(52) reads

$-\chi_{1,2}^{\prime\prime}+\left(\frac{m^{2}}{R^{2}+u^{2}}\pm
m\frac{u}{(R^{2}+u^{2})^{3/2}}\right)\chi_{1,2}=\epsilon^{2}\chi_{1,2},$ (65)

It is worthwhile to mention that the effective potential

$U^{2}_{eff1,2}=\frac{m^{2}}{R^{2}+u^{2}}\pm m\frac{u}{(R^{2}+u^{2})^{3/2}},$
(66)

couples the angular momentum quantum number $m$ and the curved geometry terms.
The second term in the potential $\frac{m}{(R^{2}+u^{2})^{3/2}}$ breaks the
symmetry $m\rightarrow-m$. Indeed, we can obtain the $\chi_{2}$ spinor
component from $\chi_{1}$ by performing the change $m\rightarrow-m$.

For $u>>R$, the potential tends to $U_{eff1,2}\approx\frac{m(m\pm 1)}{u^{2}}$.
Accordingly, the Eq.(65) has the asymptotic solution

$\chi_{1,2}\approx\sqrt{u}\left(c_{1}J_{\frac{2m\pm 1}{2}}(\epsilon
u)+c_{2}Y_{\frac{2m\pm 1}{2}}(\epsilon u)\right),$ (67)

where $J_{n}(x)$ and $Y_{n}(x)$ are the Bessel function of the first kind and
the second kind, respectively. In this region, the solution in Eq.(67)
resembles the one found for the Dirac equation in other graphene wormhole
geometries outside the throat wormhole4 .

Note the presence of the total angular momentum number $m=l+\frac{1}{2}$ in
the order of the Bessel functions. For $u\rightarrow\pm\infty$, the potential
vanishes and thus, the $\chi_{1}$ function tends to $A\sin(\epsilon u)$, as
for the $m=0$ states. Therefore, the interaction between angular momentum and
curvature is concentrated around the graphene wormhole throat.

For $R=1$ and $\epsilon=1$, we numerically solved the Eq.(65) and the
resulting squared wave function was plotted in fig.(15) for $m=1$ and in the
fig.(15) for $m=-1$. By changing $m\rightarrow-m$, the density of state is
shift from the upper $(u>0)$ into the lower sheet $(u<0)$. Moreover, for
$\beta=1$ (thick line) the amplitude is smaller than for $\beta=0.5$ (dashed
line).

The zero mode, i.e., for $\epsilon=0$ is given by

$\psi^{0}_{1,2}(u)=(R^{2}+u^{2})^{-1/4}e^{-\bar{\beta}\sigma(u)}e^{\mp
m\sinh^{-1}(u/R)},$ (68)

where is evident the chiral symmetry breaking $m\rightarrow-m$ and the parity-
odd behaviour of this ground state. Indeed, as shown in the fig. the states
for $m>0$ are suppressed in the upper layer whereas the $m<0$ states are
suppressed in the lower layer. A similar chiral separation was also found for
the massless Dirac field in a helicoidal graphene strip atanasov .

Figure 16: Zero mode for $m=\pm 1/2$.

Figure 17: Zero mode $m=\pm 3/2$.

###  V.2 Constant magnetic field

Now let us consider the additional effects from the uniform magnetic field.
The respective Klein-Gordon like equation becomes

$-\chi_{1,2}^{\prime\prime}+\left(\frac{(m+(Bu^{2})/2)^{2}}{R^{2}+u^{2}}\mp\frac{u(-2m+B(2R^{2}+u^{2}))}{2(R^{2}+u^{2})^{3/2}}\right)\chi_{1,2}=\epsilon^{2}\chi_{1,2}.$
(69)

In the eq.(69), the effective potential

$U^{2}_{eff1,2}=\frac{(m+(Bu^{2})/2)^{2}}{R^{2}+u^{2}}\mp\frac{u(-2m+B(2R^{2}+u^{2}))}{2(R^{2}+u^{2})^{3/2}}$
(70)

includes effects from the curved geometry, total angular momentum (spin and
orbital), and the coupling to the magnetic field. Note that, for $u\gg R$
(outside the throat), the effective potential takes the form

$U^{2}_{eff}\approx\frac{m(m+1)}{u^{2}}+B\left(m-\frac{1}{2}\right)+\frac{B^{2}}{4}u^{2},$
(71)

which is the effective potential for a $2+1$ massless Dirac fermion under an
uniform magnetic field in a flat plane using cylindrical coordinates
diracplanar ; diracplanar2 . That is an expected result, since the graphene
wormhole surface is asymptotic flat. Moreover, the effective potential in
Eq.(70) also exhibits the $m\rightarrow-m$ asymmetry.

Due to the complexity of eq.(69) we employ numerical methods to obtain the
first excited states and their respective energy spectrum (Landau levels). In
the figures (19) and (19) we plotted the effective potential $U^{2}_{eff}$ for
$m=0$ and $m=1$, respectively. Note that for $u\gg R$, the effective potential
diverges as $\frac{B^{2}}{4}u^{2}$, whereas for $u\ll R$ the potential is
dominated by the geometric and angular momentum terms (finite barrier for
$m\neq 0$).

We plotted the first Landau levels for $eB=1$, $R=1$, $\beta=1$ and $m=0$ (s
state) in the fig.(21. Note that the first excited state (red line) is located
on the upper layer, whereas the second (blue) and the third (green) have two
asymmetric peaks around the origin. For $eB=1$, $R=1$, $\beta=1$ and $m=2$ in
the fig.(21), the first excited state already has two asymmetric peaks
displaced from the origin. Nonetheless, it is worthwhile to mention that the
probability density does not vanishes at the origin. Note that for $u\gg R$,
the wave function exhibits the usual exponential decay due to the external
magnetic field diracplanar . Therefore, the external magnetic field allow us
to confine the electron around the wormhole throat. However, due to the curved
geometry and the strain, the electron is not symmetric confined around the
wormhole.

Figure 18: Effective squared potential for $m=-3/2$ for different values of
$B$.

Figure 19: Effective squared potential for $m=3/2$. Red line(B=0.1), blue line
(B=0.15) and the green line (B=0.2).

Figure 20: Density of states for $B=0.1$ and $m=1/2$. The ground state (red
line) has a peak in the upper layer whereas the first excited state (blue
line) is more localized in the lower layer. The second excited state exhibits
three less distinct peaks.

Figure 21: Density of states for $B=0.1$ and $m=3/2$.The ground state (red
line) has a peak in the upper layer whereas the first excited state (blue
line) is more localized in the lower layer. The second excited state exhibits
three less distinct peaks.

Finally, the ground state zero mode under the action of the magnetic field is
modified by

$\displaystyle\psi^{0}_{1,2}(u)$ $\displaystyle=$
$\displaystyle(R^{2}+u^{2})^{-1/4}e^{-\bar{\beta}\sigma(u)}e^{\mp
m\sinh^{-1}(u/R)}$ (72) $\displaystyle\times
e^{\mp\frac{eB}{4}(u\sqrt{R^{2}+u^{2}}+R^{2}\tanh^{-1}(u/\sqrt{R^{2}+u^{2}}))}.$

In the eq.(72), the magnetic field introduces another exponential factor whose
argument is a parity-odd function. Accordingly, the magnetic field enhances
the chiral separation of the eletronic modes. However, note that shifting the
sign of the magnetic field for a given angular momentum number $m$, the
magnetic field might reduce the chiral separation between the upper and lower
layers.

##  VI Final remarks and perspectives

We investigated the curvature, strain and magnetic field effects upon a
massless relativistic electron on a graphene wormhole surface. The graphene
wormhole was described by a catenoid surface which smoothly connects two
asymptotic flat graphene planes (layers). In this sense, the geometry proposed
was a smooth generalization of the graphene wormhole shown in the Ref.wormhole
. The effective Dirac Hamiltonian containing strain–dependent terms was
obtained in Ref.vozmediano  and extended in Ref.vozmediano2 .

Due to the axial symmetry, we considered an isotropic strain tensor localized
near the wormhole throat, similar to the behavior of the Gaussian curvature.
Indeed, since the curved geometry of the throat was obtained due to
deformation of the lattice structure, it was expected that both curvature and
strain were localized around this region. It turned out that the
pseudo–magnetic potential due to the strain $\Gamma_{u}$ had only components
along the meridian coordinate $u$, whereas the spin connection $\Omega_{\phi}$
pointed along the parallel direction $\phi$. In this manner, despite of having
the same origin (the lattice deformation), these two interactions had distinct
effects on the electron. Moreover, by applying the external magnetic field, a
true magnetic potential $\vec{A}$ also acted on the electron. Nevertheless,
although $\vec{A}$ only had the angular component $A_{\phi}$, the spin
connection $\Omega_{\phi}$ was parity–odd, whereas $A_{\phi}$ was parity–even
under the transformation $u\rightarrow-u$.

The strain and spin–curvature brook the parity invariance of the effective
Hamiltonian. Moreover, the spin–connection term led to a chiral invariance
$m\rightarrow-m$. By employing the supersymmetric quantum mechanichal (QMSUSY)
approach, we found that the strain vector yielded to an exponential
suppression of the wave function, whereas the spin connection led to a
power–law decay. In absence of magnetic field, the superpotential was given by
the spin–curvature term which increased the amplitude of the probability
density in the upper layer for $m>0$ and in the lower layer for $m<0$. A
similar chiral behavior was found in graphene nanoribbons in a helical shape
atanasov . Since the graphene structure is asymptotically flat, for a
vanishing strain vector, the wave function behaves as a free wave in a flat
plane watanabe . The inclusion of an uniform magnetic field confined the
electronic states near the wormhole throat. These Landau levels were modified
by the spin–curvature and the strain interactions turned out to be an
asymmetric probability distribution.

In addition, this work revealed that, in spite of the coupling of strain,
curvature and magnetic field in the effective Dirac Hamiltonian were similar,
they possessed rather different effects. As a result, we pointed out a couple
of perspectives for further investigations. A noteworthy extension of the
present work could be considering the effects of a a dynamical strain
(phonons) or corrugations on the graphene wormhole. Furthermore, the chiral
breaking due to the spin–curvature coupling suggests an interesting spin–Hall
current to be analysed. Moreover, the analysis of the geometric Aharonov–Bohm
like phase due to the concentrated curvature around the wormhole throat seems
a worthy perspective as well.

## Acknowledgments

J.E.G.Silva thanks the Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq), grants no 304120/2021-9 for financial support.
Particularly, A. A. Araújo Filho would like to thank Fundação de Apoio à
Pesquisa do Estado da Paraíba (FAPESQ) and CNPq – [200486/2022-5] and
[150891/2023-7] for the financial support. JF would like to thank the Fundação
Cearense de Apoio ao Desenvolvimento Científico e Tecnológico (FUNCAP) under
the grant PRONEM PNE0112-00085.01.00/16 for financial support, the CNPq under
the grant 304485/2023-3 and Gazi University for the kind hospitality. Most of
these calculations were performed by using the Mathematica software.

## References

  * (1) A. K. Geim, Science, v. 324, (2009) 1530.
  * (2) A. Kara et al, Surf. Sci. Rep. 67, 1, (2012).
  * (3) A. Carvalho, M. Wang, X. Zhu, A. S. Rodin, H. Su, A. H. Castro Neto, Nat. Rev. Mat. 1, 11 (2016).
  * (4) M. I. Katsnelson, Graphene: carbon in two dimensions, Cambridge University Press (2012).
  * (5) K. S. Novoselov, K. Geim, V. Morozov, D. Jiangy, Y. Zhangs, S. V. Dubonos, V. Grigorieva and A. A. Firsov, Science 306, (2004) 666.
  * (6) A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
  * (7) M. I. Katsnelson, Eur. Phys. J. B 51, 157–160 (2006).
  * (8) M. I. Katsnelson, K. S. Novoselov, A. K. Geim, Nature Physics volume 2, 620 (2006).
  * (9) A. V. Shytov, M. I. Katsnelson, L. S. Levitov, Phys. Rev. Let. 99, (2007).
  * (10) A. Iorio, G. Lambiase, Phys. Lett. B 716, 334 (2012).
  * (11) T. Morresi et al, 2D Mater. 7, 041006 (2020).
  * (12) M. Burgess, B. Jensen, Phys. Rev. A 48, 3, 1861 (1993).
  * (13) F.T. Brandt, J.A. Sanchez-Monroy, Phys. Let. A 380, 38, 3036 (2016).
  * (14) H. Zhao, Y. L. Wang, C.Z. Ye, R. C., G. H. Liang, and H. Liu Phys. Rev. A 105, 052220 (2022).
  * (15) Q. H. Liu, Z. Li, X. Y. Zhou, Z. Q. Yang, W. K. Du Eur. Phys. J. C 79, 712 (2019).
  * (16) J. González, F. Guinea, M.A.H. Vozmediano, Phys. Rev. Lett. 69, 172 (1992).
  * (17) J. González, F. Guinea, M.A.H. Vozmediano, Nuclear Physics B, (1993).
  * (18) Saito R., Dresselhaus G. and Dresselhaus M. S., Physical Properties of Carbon Nanotubes, I. C. Press, London 1998.
  * (19) P. Lammert and V. Crespi, Phys. Rev. lett. 85, 5190 (2000).
  * (20) F. de Juan, A. Cortijo, M. A. H. Vozmediano, Phys. Rev. B 76, 165409, (2007).
  * (21) C. Furtado , F. Moraes , A.M. de M. Carvalho, Phys. Lett. A 372, 5368, (2008).
  * (22) V. Atanasov, A. Saxena, Phys. Rev. B 92, 035440 (2015).
  * (23) M. Watanabe, H. Komatsu, N. Tsuji, and H. Aoki Phys. Rev. B 92, 205425 (2015).
  * (24) V. Atanasov, A. Saxena, Phys. Rev. B 81, 205409 (2010).
  * (25) K. Flouris, M. M. Jimenez, and H. J. Herrmann, Phys. Rev. B 105, (2022) 235122.
  * (26) L. N. Monteiro, C. A. S. Almeida, and J. E. G. Silva Phys. Rev. B 108, 115436 (2023).
  * (27) Ö. Yeşiltaş Advances in high energy physics, 6891402 (2018).
  * (28) Ö. Yeşiltaş and J. Furtado, Int. J. Mod. Phys. A 37 (2022) no.11n12, 2250073
  * (29) F. de Juan, A. Cortijo, María A. H. Vozmediano, A. Cano, Nature Physics 7, 810 (2011).
  * (30) A. Shitade and E. Minamitani, New J. Phys. 22, 113023 (2020).
  * (31) G. Liang, Y. Wang, M. Lai, H. Liu, H. Zong and S. Zhu Phys. Rev. A 98, 062112 (2018).
  * (32) Y. Wang, H. Zhao, H. Jiang, H. Liu and Y. Chen Phys. Rev. B 106, 235403, (2022).
  * (33) F. Guinea, A. K. Geim, M. I. Katsnelson, and K. S. Novoselov, Phys. Rev. B 81, 035408 (2010).
  * (34) M.A.H. Vozmedianoa, M.I. Katsnelson, F. Guinea, Phys. Rep. 496, 109 (2010).
  * (35) Levy, N. et al., Science 329, 544–547 (2010).
  * (36) Guinea, F., Katsnelson, M. I. and Geim, A. K., Nature Phys. 6, 30–33 (2010).
  * (37) F. de Juan, J. L. Mañes, and M. A. H. Vozmediano Phys. Rev. B 87, 165131 (2013).
  * (38) J. L. Mañes, F. de Juan, M. Sturla, and M. A. H. Vozmediano, Phys. Rev. B 88, 155405 (2013).
  * (39) A. Sinner and K. Ziegler, Ann. Phys. 400, 262-278 (2019).
  * (40) E. Arias, A. R. Hernandez and C. Lewenkopf, Phys. Rev. B 92, 245110 (2015).
  * (41) J. González, J. Herrero, Nucl. Phys. B 825, 426 (2010).
  * (42) J. Gonzalez, F. Guinea, J. Herrero, Phys. Rev. B 79, 165434 (2009).
  * (43) R. Pincak, J. Smotlacha, Quantum Matter 5, 114 (2016).
  * (44) G.Q. Garcia, P.J. Porfírio, D.C. Moreira and C. Furtado, Nucl. Phys. B 950, 114853 (2020).
  * (45) R. Dandoloff, A. Saxena, B. Jensen, Phys. Rev. A 81, 014102 (2010).
  * (46) J.E.G. Silva et al., Phys. Lett. A 384, 126458 (2020).
  * (47) T. F. de Souza, A. C. A. Ramos, R. N. Costa Filho and J. Furtado, Phys. Rev. B 106 (2022) no.16, 165426
  * (48) K. Pimsamarn, P. Burikham, T. Rojjanason, Eur. Phys. J. C 80, 1111 (2020).
  * (49) Ö. Yeşiltaş, J. Furtado, and J. E. G. Silva, Eur. Phys. J. Plus, 137, (2022) 416.
  * (50) E. Cavalcante, Eur. Phys. J. Plus 137, 1351 (2022).
  * (51) C. Ho and V. R. Khalilov, Phys. Rev. A 61, 032104 (2000).
  * (52) V.M. Villalba and A. R. Maggiolo, Eur. Phys. J. B 22, 31–35 (2001).
  * (53) M. J. Bueno, C. Furtado and A. M. de M. Carvalho, Eur. J. Phys. B 85, 53 (2012).


# On the complexity of open shop scheduling with time lags

Wiesław Kubiak

_Faculty of Business Administration_
_Memorial University_
_St. John’s, Canada_

###### Abstract

The minimization of makespan in open shop with time lags has been shown NP-
hard in the strong sense even for the case of two machines and unit-time
operations. The minimization of total completion time however has remained
open for that case though it has been shown NP-hard in the strong sense for
weighted total completion time or for jobs with release dates. This note shows
that the minimization of total completion time for two machines and unit-time
operations is NP-hard in the strong sense which answers the long standing open
question.

Keywords: Open shop, time lags, total completion time, complexity

##  1 Introduction

The open shop with job-dependent time lags has been studied for quite sometime
in the literature. The time lags model delays required between job’s
operations due to necessary transportation needed to move a job from one
machine to another for instance. Zhang [7] provides an interesting discussion
of the time lag models and their applications in scheduling. Most research on
the open shops with time lags has focused on two machine open shops where each
job $J_{i}$, $i=1,\dots,n$, consists of two operations $O_{i1}$ and $O_{i2}$
to be processed on two machines $M_{1}$ and $M_{2}$ respectively in any order.
The operations $O_{i1}$ and $O_{i2}$ processing times equal $p_{i1}>0$ and
$p_{i2}>0$ respectively, and the time lag is $l_{i}\geq 0$. In a feasible
schedule either machine can process at most one job at a time, each job can be
processed by at most one machine at a time, and the _later_ operation of job
$J_{i}$ in the schedule needs to wait at least $l_{i}$ time units to start
following the completion of the _earlier_ operation of job $J_{i}$ in the
schedule. Yu [5], and Yu et al. [6] prove a series of strong complexity
results for makespan minimization. They prove that the problem is NP-hard in
the strong sense even if all operations are unit-time operations. This problem
is denoted by $O2|p_{ij}=1,l_{i}|C_{\max}$ in the well-known notation of
Graham et al [2]. Yu [5] then goes on to prove that the problem is NP-hard in
the strong sense even if there are only two possible values $l$ and
$l^{\prime}$ of time lags in a job-proportionate open shop, i.e. the problem
$O2|p_{i1}=p_{i2},l_{i}\in\\{l,l^{\prime}\\}|C_{max}$, and it is also NP-hard
in the ordinary sense when only one value $l$ of time lag is permitted in a
job-proportionate open shop, i.e. the problem
$O2|p_{i1}=p_{i2},l_{i}=l|C_{max}$. Rebain and Strusevich [3] give a linear
time algorithm for the instances with short time lags, i.e time lags that meet
the following condition $\max_{i}\\{l_{i}\\}\leq\min_{ij}\\{p_{ij}\\}$. These
results determine current borderline between NP-hard and polynomially solvable
cases for the two machine open shop makespan minimization problem with job
dependent-time lags. The problem intractability caused research to focus on
approximation algorithms and on-line competitive algorithms for makespan
minimization. Strusevich [4] gave $\frac{3}{2}$\- approximation algorithm, and
Zhang and van de Velde [8] gave a 2-competitive algorithm for
$O2|l_{i}|C_{max}$. The reader is referred to Zhang [7] for a comprehensive
review of approximation and on-line algorithms for the problem

Brucker et al. [1] switch attention to other than makespan objective
functions. In particular to the total completion time objective which is
another key scheduling objective function. They prove that _weighted_ total
completion time minimization is NP-hard in the strong sense, i.e. the problem
$O2|p_{ij}=1,l_{i}|\sum w_{i}C_{i}$. They prove that the same holds for the
total completion time with jobs being _released_ possibly at different times,
i.e. the problem $O2|p_{ij}=1,l_{i},r_{i}|\sum C_{i}$. In this paper we prove
that the problem where all jobs are released at the same time and their
weights are all equal, i.e. the problem $O2|p_{ij}=1,l_{i}|\sum C_{i}$ is NP-
hard in the strong sense. This result strengthens those earlier complexity
results for total completion time, and it answers a question that has been
open at least since the paper by Brucker et al. [1]. The prove is given in the
next section.

##  2 NP-hardness proof

Let $l_{1},\dots,l_{n},e$ be non-negative integers, and $n$ a positive _even_
integer such that $n<e<\frac{3n}{2}$. Let $A$ and $B$ be a partition of the
index set $\\{1,\dots,n\\}$ into two disjoint sets of equal size
$\frac{n}{2}$. For simplicity denote
$\\{\ell_{1},\dots,\ell_{\frac{n}{2}}\\}=\\{l_{j}:j\in A\\}$ and
$\\{\lambda_{1},\dots,\lambda_{\frac{n}{2}}\\}=\\{l_{j}:j\in B\\}$. Consider
the following question.

(Q) Is there a partition of the set $\\{1,\dots,n\\}$ into two disjoint sets
$A$ and $B$ of equal size $\frac{n}{2}$ such that there is a permutation
$\pi_{A}$ of the set $\\{1,\dots,\frac{n}{2}\\}$ and a permutation
$\sigma_{A}$ of the set $\\{\frac{n}{2}+1,\dots,n\\}$ satisfying

$\pi_{A}(i)+\ell_{i}+\sigma_{A}(i)=e$ (1)

for $i=1,\dots,\frac{n}{2}$, and there is a permutation $\pi_{B}$ of the set
$\\{\frac{n}{2}+1,\dots,n\\}$ and a permutation $\sigma_{B}$ of the set
$\\{1,\dots,\frac{n}{2}\\}$ satisfying

$\pi_{B}(i)+\lambda_{i}+\pi_{A}(i)=e$ (2)

for $i=1,\dots,\frac{n}{2}$?

We refer to this problem as Partition Restricted Numerical 3-Dimensional
Matching (PRN3DM) problem. It is easy to verify that any instance of the
PRN3DM problem with an affirmative answer to Q must satisfy the following
condition

$\Sigma_{i=1}^{n}l_{i}=n(e-n-1),$ (3)

therefore without loss of generality we limit the PRN3DM to the instances for
which (3) holds. The problem PRN3DM is NP-hard in the strong sense which
follows from Theorem 1 on p. 30 in Yu [5].

The corresponding instance of the decision open shop problem
$O2|p_{ij}=1,l_{i}|\Sigma C_{i}\leq F$ is made up of $n$ jobs
$J_{1},\dots,J_{n}$ with time lags
$L_{1}=l_{1}+\Delta,\dots,L_{n}=l_{n}+\Delta$ respectively, where
$\Delta=\frac{3n}{2}-e$. The threshold for total completion time equals
$F=\frac{n}{2}(\frac{3n}{2}+1)$.

For a partition $A$ and $B$ and permutations $\pi_{A}$, $\sigma_{A}$,
$\pi_{B}$, and $\sigma_{B}$ that attest to an affirmative answer to Q, the
schedule $S$ in Figure 1 is a feasible schedule for the open shop problem with
total completion time equal to

$\Sigma_{i=1}^{\frac{n}{2}}(\pi_{A}(i)+\ell_{i}+\Delta+1)+\Sigma_{i=1}^{\frac{n}{2}}(\pi_{B}(i)-\frac{n}{2}+\lambda_{i}+\Delta+1),$

which by (3) and definition of $\Delta$ equals $F$. Thus $S$ gives an
affirmative answer to the problem $O2|p_{ij}=1,l_{i}|\Sigma C_{i}\leq F$
instance.

Figure 1: Schedule S for the partition $A$ and $B$ and permutations $\pi_{A}$,
$\sigma_{A}$, $\pi_{B}$, and $\sigma_{B}$.

Now, let $\mathcal{S}$ be a feasible schedule for the instance of
$O2|p_{ij}=1,l_{i}|\Sigma C_{i}\leq F$ with total completion time not
exceeding $F$. We first show that the makespan $C_{\max}=n$ in $\mathcal{S}$.
To that end let $x_{\sigma(1)}\leq x_{\sigma(2)}\leq\dots\leq
x_{\sigma(n-1)}\leq x_{\sigma(n)}$ be the times when the _earlier_ operations
of the jobs $J_{1},\dots,J_{n}$ complete in $\mathcal{S}$. Because of the
delay due to the time lags the total completion time of $\mathcal{S}$ is at
least

$\sum_{i=1}^{\frac{n}{2}}(x_{\sigma(2i-1)}+x_{\sigma(2i)})+\sum_{j=1}^{n}(L_{j}+1),$

which does not exceed the threshold $F$ for $\mathcal{S}$. Hence by (3) and
definition of $\Delta$

$\sum_{i=1}^{\frac{n}{2}}(x_{\sigma(2i-1)}+x_{\sigma(2i)})\leq\frac{n}{2}(\frac{n}{2}+1).$
(4)

For two machines we have $i\leq x_{\sigma(2i-1)}\leq x_{\sigma(2i)}$,
$i=1,\dots,\frac{n}{2}$. Thus by (4) we get
$x_{\sigma(2i-1)}=x_{\sigma(2i)}=i$ for $i=1,\dots,\frac{n}{2}$. Therefore
each job $J_{1},\dots,J_{n}$ completes after time $\frac{n}{2}$ in
$\mathcal{S}$. Let $C_{\pi(1)}\leq C_{\pi(2)}\leq\dots\leq C_{\pi(n-1)}\leq
C_{\pi(n)}$ be the completion times of the jobs $J_{1},\dots,J_{n}$ in
$\mathcal{S}$. Clearly $C_{\pi(i)}=\frac{n}{2}+c_{\pi(i)}$, for some
$c_{\pi(i)}\geq 1$, thus

$\sum_{i=1}^{\frac{n}{2}}(c_{\pi(2i-1)}+c_{\pi(2i)})\leq\frac{n}{2}(\frac{n}{2}+1)$
(5)

in $\mathcal{S}$. Again, for two machines we have $i\leq c_{\pi(2i-1)}\leq
c_{\pi(2i)}$, $i=1,\dots,n$. Thus by (5) we get $c_{\pi(2i-1)}=c_{\pi(2i)}=i$
for $i=1,\dots,\frac{n}{2}$. Therefore all jobs complete by $C_{\max}=n$ in
$\mathcal{S}$ which is what we set out to show first.

Figure 2: Schedule S with total completion time not exceeding F.

Finally, let $C$ be the set of $\frac{n}{2}$ jobs with earlier operations in
the interval $[0,\frac{n}{2}]$ on $M_{1}$ and later operations in
$[\frac{n}{2},n]$ on $M_{2}$ in $\mathcal{S}$, and $D$ be the set of
$\frac{n}{2}$ jobs with earlier operations in the interval $[0,\frac{n}{2}]$
on $M_{2}$ and later operations in $[\frac{n}{2},n]$ on $M_{1}$ in
$\mathcal{S}$, see Figure 2. We have

$\alpha_{C}(i)+L^{\prime}_{i}+\beta_{C}(i)=n$

for each $i\in C$, and

$\alpha_{D}(j)+L^{\prime}_{j}+\beta_{D}(j)=n$

for each $j\in D$ for some permutations $\alpha_{C}$, $\alpha_{D}$,
$\beta_{C}$, $\beta_{D}$ of the set $\\{1,\dots,\frac{n}{2}\\}$. Thus

$\alpha_{C}(i)+L^{\prime}_{i}+\beta_{C}(i)+\frac{n}{2}=\frac{3n}{2}$

for each $i\in C$, and

$\alpha_{D}(j)+\frac{n}{2}+L^{\prime}_{j}+\beta_{D}(j)=\frac{3n}{2}$

for each $j\in D$. By taking the permutation $\pi_{C}=\alpha_{C}$ of
$\\{1,\dots,\frac{n}{2}\\}$ and $\sigma_{C}=\beta_{C}+\frac{n}{2}$ of
$\\{\frac{n}{2}+1,\dots,n\\}$, and the permutation
$\pi_{D}=\alpha_{D}+\frac{n}{2}$ of $\\{\frac{n}{2}+1,\dots,n\\}$ and
$\sigma_{D}=\beta_{D}$ of $\\{1,\dots,\frac{n}{2}\\}$, we get

$\pi_{C}(i)+l^{\prime}_{i}+\sigma_{C}(i)=\frac{3n}{2}-\Delta=e$ (6)

for each $i\in C$, and

$\pi_{D}(j)+l^{\prime}_{j}+\sigma_{D}(j)=\frac{3n}{2}-\Delta=e$ (7)

for each $j\in D$, where $L^{\prime}_{i}=l^{\prime}_{i}+\Delta$ and
$l^{\prime}_{i}\geq l_{i}$ for $i=1,\dots,n$. By (6) and (7) we have

$\Sigma_{i=1}^{n}l^{\prime}_{i}=n(e-n-1).$

Thus by (3), $l^{\prime}_{i}=l_{i}$ for $i=1,\dots,n$. Hence

$\pi_{C}(i)+l_{i}+\sigma_{C}(i)=e$ (8)

for each $i\in C$, and

$\pi_{D}(j)+l_{j}+\sigma_{D}(j)=e$ (9)

for each $j\in D$. Therefor $C$ and $D$ make up the required partition, which
proves the following theorem.

######  Theorem 2.1.

The problem $O2|p_{ij}=1,l_{i}|\sum C_{i}$ is NP-hard in the strong sense.

## References

  * [1] P. Brucker, S. Knust, T. C. E. Cheng, and N. V. Shakhlevich.  Complexity results for flow-shop and open-shop scheduling problems with transportation delays.  Annals of Operations Research, 129:81–106, 2004.
  * [2] R. L. Graham, E. L. Lawler, J. K. Lenstra, and A. H. G. Rinnooy Kan.  Optimization and approximation in deterministic sequencing and scheduling: A survey.  Anns. Discr. Math., 5:287–326, 1979.
  * [3] D. Rebaine and V.A. Strusevich.  Two-machine open shop scheduling with special transportation times.  Journal of the Operational Research Society, 50:756 – 764, 1999\.
  * [4] V.A. Strusevich.  A heuristic for the two-machine open-shop scheduling problem with transportation times.  Discrete Appl. Math., 93:287 – 304, 1999.
  * [5] W. Yu.  The Two-machine Flow Shop Problem with Delays and the One- machine Total Tardiness Problem.  PhD thesis, Eindhoven University of Technology, 1996.
  * [6] W. Yu, J. A. Hoogeveen, and J. K. Lenstra.  Minimizing makespan in a two-machine flow shop with delays and unit-time operations is np-hard.  Journal of Scheduling, 7:333–348, 2004.
  * [7] X. Zhang.  Scheduling with Time Lags.  PhD thesis, Erasmus Universiteit Rotterdam, 2010.
  * [8] X. Zhang and S. van de Velde.  On-line two-machine open shop scheduling with time lags.  European Journal of Operational Research, 204:14–15, 2010.