Dirac equation on a catenoid bridge: a supersymmetric approach

The geometry of the graphene layer geim ; novoselov ; katsnelson plays an important role in the electronic structure. The possibility of developing new electronic devices due to the effect of graphene’s curvature is promoting a growing interest in quantum mechanics at curved surfaces. Electronic properties of a graphene nanotorus were studied considering the dynamics governed by the Schrödinger equation GomesSilva:2020fxo , dainamle as well as by Dirac equation Yesiltas:2018zoy , dainam2 . Geometry fluctuations produce pseudomagnetic fields ribbons whose effects can be seen at ripples contijo and corrugated layers corrugated . The curvature at the tip of a conical layer produces a topological phase furtado , whereas helical strips induce chiral properties dandoloff1 ; atanasovhelicoid ; atanasov .

Another interesting geometry studied in the last years is the catenoid geometry. As a minimal surface, two dimensional wormhole geometry is equivalent to a catenoid which are possible solutions for traversable wormholes Wo . In Ref. wormhole ; picak a bridge connecting a bilayer graphene was proposed using a nanotube. In order to obtain a smooth bridge, Ref.Dandoloff , dandoloff suggested a catenoid surface to describe the bilayer and the bridge using only one surface. This can be achieved due to the catenoid curvature which is concentrated around the bridge and vanishes asymptotically spivak . For a non-relativistic electron, the surface curvature induces a geometric potential in the Schrödinger equation. The effects of the geometry and external electric and magnetic fields upon the graphene catenoid bridge was explored in Ref.euclides , where a single electron is governed by the Schrödinger equation on the surface. Incidentally, the influence of a position-dependent mass problem upon the electron on a catenoid bridge was studied in Ref. euclides2 , where it was proposed an isotropic position-dependent mass as a function of the Gaussian and mean curvatures.

The supersymmetry (SUSY) arose as an effort to obtain an unified description of all interactions of nature wess , but it was immediately realized that it provides powerful techniques to obtain analytical solutions for problems in several branches of physics s1 . The factorization method was first introduced by Dirac in order to obtain the spectrum of the quantum harmonic oscillator dirac . Later, Schrödinger applied the same method for the radial part of the Coulomb problem schrodinger . The supersymmetric quantum mechanics (SUSY QM) is an intense field of research and nowadays there is a wide set of Hamiltonians which are analytically solvable through the factorization method sukhatme . On the other hand, there are more representations of SUSY QM such as shape invariance R , Darboux transformations F and Hamilton hierarchies DJ . SUSY QM has received lot of interest for beautiful mathematical insight in both relativistic and non-relativistic quantum mechanics 1 ; 2 ; 3 .

In this paper we study some properties of a relativistic electron constrained to move on a catenoid surface. Unlike the non-relativistic electron, the Dirac electron does not couple directly to the surface curvature. Instead, the momentum operator (covariant derivative) is modified by the surface tetrads and connection 1-forms. In section II, we study the massless Dirac equation on the catenoid and we decoupled the two components of the spinor. As a result, we obtain two second-order Klein-Gordon-like equations with an effective potential depending on the surface curvature and on the effective Fermi velocity. In section III we present briefly the SUSY QM approach which was used to find analytical solutions for the Dirac equation on the catenoid in two particular cases, namely, the constant Fermi velocity case and the position-dependent Fermi velocity case. In section IV we present our final remarks.

The catenoid can be described by the meridian $u\in(-\infty,\infty)$ and the parallel $\phi\in[0,2\pi)$, as

where $R$ is the radius of the catenoid bridge and $u=u(z)=R\sinh(z/R)$ , as depicted in Fig.(1). Using this coordinate system, the $2+1$ spacetime interval reads

Consider an orthogonal moving frame defined by the tangent vectors $\hat{e}_{1}=\frac{\partial\vec{r}}{\partial u}$, $\hat{e}_{2}=\frac{1}{\sqrt{R^{2}+u^{2}}}\frac{\partial\vec{r}}{\partial\phi}$ and the normal unit vector $\hat{e}_{3}=\hat{e}_{1}\times\hat{e}_{2}$. The respective co-frame is given by $\omega^{0}=dt$, $\omega^{1}=du$, $\omega^{2}=\sqrt{R^{2}+u^{2}}d\phi$ and $\omega^{3}=0$, where $\omega^{a}=e^{a}_{\mu}dx^{\mu}$. Thus, $d\vec{r}=\omega^{1}\otimes\hat{e}_{1}+\omega^{2}\otimes\hat{e}_{2}$ and the metric can be rewritten as $ds^{2}=\eta_{ab}\omega^{a}\otimes\omega^{b}$, being $\eta_{ab}=diag(+1,-1,-1)$. Using the torsion-free Cartan equation $T^{a}=d\omega^{a}+\omega^{a}_{b}\wedge\omega^{b}=0$ leads to the one-form connections $\omega^{a}_{b}=\Gamma_{cb}^{a}\omega^{c}$ of form $\omega^{1}_{3}=\frac{R}{R^{2}+u^{2}}\omega^{1}$, $\omega^{2}_{3}=-\frac{R}{R^{2}+u^{2}}\omega^{2}$ and $\omega^{2}_{1}=\frac{u}{R^{2}+u^{2}}\omega^{2}$. Since $d\hat{e}_{3}=\omega^{1}_{3}\vec{e}_{1}+\omega^{2}_{3}\vec{e}_{2}=K_{b}^{a}\omega^{a}\otimes\vec{e}_{b}$, then the second fundamental form $K^{a}_{b}$ has components

Therefore, the catenoid has gaussian and mean curvatures

Accordingly, the catenoid has negative gaussian curvature vanishing far from the the bridge throat.

Intrinsically, only the $\omega^{2}_{1}$ is non zero. Using the metric compactibility condition $\omega_{ab}+\omega_{ba}=0$ and using $\omega^{12}=\omega^{12}_{\mu}dx^{\mu}$, then

The massless Dirac equation on a surface reads

where $v_{F}$ is the Fermi velocity. The covariant spinor derivative is $D_{\mu}=\partial_{\mu}-\Gamma_{\mu}$, where $\Gamma_{\mu}=\frac{1}{4}\Gamma^{ab}_{\mu}\gamma_{a}\gamma_{b}$ is the spinor connection. From Eq.(7) the only non-vanishing spinor connection component is $\Gamma_{\phi}=\frac{1}{4}\Gamma^{12}_{\phi}\gamma_{1}\gamma_{2}$. In (2+1)-D we can adopt the following representation for the flat $\gamma^{a}$ matrices $\gamma_{0}=\sigma_{3}$, $\gamma_{1}=-i\sigma_{2}$ and $\gamma_{2}=-i\sigma_{1}$. The curved $\gamma^{\mu}$ matrices are related to the flat $\gamma^{a}$ matrices by $\gamma^{\mu}=e^{\mu}_{a}\gamma^{a}$, so that,

and the spinor connection is

The massless Dirac equation takes the form

Therefore, the catenoid geometry leads to a geometric potential as $V_{1}(u)=\frac{u}{4(R^{2}+u^{2})}\gamma_{1}$, which vanishes asymptotically and exhibits a parity odd behaviour near the throat. The Dirac equation can be written as

where the Hamiltonian is given by

The independence of the catenoid metric with respect to time and the angular coordinate $\phi$ leads to the conservation of energy and of the angular momentum along the direction $z$. Thus, the wave function can be written as $\Psi(t,u,\phi)=e^{i\frac{E}{\hbar}t}e^{im\phi}\psi(u)$ what leads to the Dirac hamiltonian

Using the Hamiltonian in eq.(18), the Dirac equation leads to coupled equations for the two components of the spinor $\psi=\left(\begin{array}[]{cc}\psi_{1}\\ \psi_{2}\end{array}\right)$, as

Decoupling the system of first-order equations in eq. (21) leads to

where $U_{1}=\left(\frac{u}{R^{2}+u^{2}}\right)^{\prime}-\left(\frac{u}{R^{2}+u^{2}}\right)^{2}+\left(\frac{m}{\sqrt{R^{2}+u^{2}}}\right)^{2}-\left(\frac{m}{\sqrt{R^{2}+u^{2}}}\right)^{\prime}$ and the prime stands for the derivative with respect to $u$. Employing the change on the wave function of the form $\psi_{1}=(R^{2}+u^{2})^{1/4}\chi_{1}$ we obtain

where the effective potential $V_{eff,1}$ is given by

Analogously for the second component of the spinor we have:

where the potential $V_{eff,2}(u)$ is exactly like $V_{eff,1}(u)$ but under the interchanging $m\rightarrow-m$. The above equations were obtained under the consideration of a constant Fermi velocity $v_{F}$. The position dependent Fermi velocity case will be addressed in the following subsection.

Let us give a brief review of some aspects of supersymmetric quantum mechanics (SUSY QM) junker that we use in our search of exact solutions for the effective potential models given in the previous Sections. The techniques based on the factorization method which takes place in SUSY QM helps to identify the Hamiltonians with solvable potentials. First, considering the one dimensional eigenvalue equation below

where $U(z)$ is a real scalar function. If we find two operators which are adjoint of each other and first order differental operators as

where the superpotential $W(z)$ is a real function and element of the first order operators. Using the product of these operators, we can introduce two partner Hamiltonians defined by

We can call the eigenfunctions of the Hamiltonians $\emph{H}_{1}$, $\emph{H}_{2}$ as $\psi^{(1)}_{n}$ and $\psi^{(2)}_{n}$ while eigenvalues as $E^{(1)}_{n}$ and $E^{(2)}_{n}$ respectively. Then, one can introduce the partner potentials in terms of superpotentials as

These partner Hamiltonians are sharing the same energy eigenvalues except the groundstate. In case of unbroken SUSY, the ground state is not degenerate with zero energy $E^{(1)}_{0}$ which also leads to

One can express the groundstate wavefunction in terms of the superpotential using (38)

When it comes to the discrete spectrum of $H_{2}$, the relationship between the energies of these Hamiltonians are given by

The eigenfunctions of the partner Hamiltonians are linked as below

We remind that if $\psi^{(1)}_{n+1}$ is normalized, then, $\psi^{(2)}_{n}$ is also normalized(non-relativistic case). The annihilation of the groundstate of the operator $H_{1}$ means that there will be no SUSY partner for this state only sukhatme .