⚛️ quantum physics

Generalized Ellis-Bronnikov graphene wormhole

This paper investigates the spinless stationary Schrödinger equation for an electron bound to a generalized Ellis-Bronnikov graphene wormhole-like surface, analyzing how the geometry-controlled parameter $n$ and orbital angular momentum influence the resulting geometric potential, bound states, and probability density.

Original authors: T. F. de Souza, A. C. A. Ramos, R. N. Costa Filho, J. Furtado

Published 2026-07-28

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CC BY 4.0

Original authors: T. F. de Souza, A. C. A. Ramos, R. N. Costa Filho, J. Furtado

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe not just as a vast, empty stage, but as a flexible fabric that can be stretched, twisted, and folded. In the world of theoretical physics, scientists have long dreamed of "wormholes"—tunnels that could connect two distant points in space, or even two different universes, acting like a cosmic shortcut. While these structures are usually imagined as massive, gravity-warping tunnels in space, there is a fascinating branch of physics that studies how similar shapes can appear in the microscopic world of materials. Specifically, scientists look at "graphene," a super-thin sheet of carbon atoms that is only one atom thick. Because graphene is so flexible, it can be bent into curves, ripples, and even shapes that look like tiny tunnels. When electrons (the tiny particles that carry electricity) move across these curved surfaces, the shape of the road changes how they behave. It's as if the electron is a car driving on a track that suddenly dips into a valley or rises over a hill; the curve of the track itself creates a "force" that pushes or pulls the car, even without any engine or brakes. Understanding how these tiny particles dance on curved surfaces helps us design new, smarter electronic devices and gives us a way to test big ideas about the universe using a sheet of carbon on a lab bench.

This paper takes that idea and asks a specific question: What happens if we shape a piece of graphene into a very specific, mathematically defined tunnel called a "generalized Ellis-Bronnikov wormhole"? The authors, working with the equations that describe how electrons move, simulated what would happen if an electron were permanently stuck to this curved graphene surface. They found that the shape of the tunnel acts like a complex landscape of hills and valleys for the electron. The key to this landscape is a number called $n$ , which controls how "cylindrical" or "tunnel-like" the shape is. When $n$ is small, the shape looks like a classic hourglass (a catenoid). As $n$ gets bigger, the tunnel straightens out, looking more like a cylinder with curved ends.

The researchers discovered that the electron doesn't just wander freely; it gets trapped in specific "bound states," which are like the electron getting stuck in a valley. When the electron has no spin or orbital twist (a state called $m=0$ ), the shape of the tunnel creates a single deep valley in the middle for a simple hourglass shape ( $n=2$ ). However, as the shape becomes more cylindrical (with $n$ values like 10, 20, or 40), that single valley splits into two deep valleys, one at each end of the tunnel. The electron can then exist in a "hybrid" state, where it is effectively in both valleys at once, or it can tunnel back and forth between them. The paper calculates that for a tunnel with a radius of 70 Å (a very small unit of length) and a specific shape ( $n=40$ ), the electron could oscillate between these two ends at a frequency of about 3 THz (terahertz). This is a very fast vibration, happening trillions of times per second.

The story gets even more interesting when the electron has "orbital angular momentum" (a kind of spin or twist, where $m=1$ ). In this case, the rules of the game change. The repulsive forces from the surface become stronger, turning the deep valleys into a high hill in the middle. For the electron to get trapped in this scenario, the tunnel needs to be very specific in its shape (requiring $n$ to be at least 18). When these conditions are met, the electron again finds two valleys at the ends of the tunnel, separated by a high barrier in the middle. The authors found that for these twisted electrons, the oscillation frequency is slightly lower, around 2.7 THz for the same size tunnel.

The paper also looked at what happens if you change the size of the tunnel's opening (the radius $R$ ). They found that making the tunnel narrower (smaller $R$ ) makes the valleys deeper, trapping the electron more tightly and changing the energy levels, while making it wider has the opposite effect. However, the basic shape of the landscape—whether it's one valley or two—doesn't change just because the size changes; it's the shape parameter $n$ that controls that. The authors conclude that by tuning these geometric parameters, we could potentially control how electrons move and vibrate in these graphene structures, creating a system where the electron acts like a tiny, oscillating clock. While this is all based on mathematical simulations and theoretical models rather than a physical experiment with a real graphene wormhole, the results suggest that the geometry of a material is just as powerful as electric or magnetic fields in controlling the behavior of electrons.

Technical Summary: Generalized Ellis-Bronnikov Graphene Wormhole

Problem Statement
This paper investigates the quantum dynamics of a spinless electron permanently bound to a generalized Ellis-Bronnikov (GEB) graphene wormhole-like surface. While the electronic properties of two-dimensional nanostructures like graphene are known to be highly dependent on geometry, and curvature induces geometric potentials affecting electron dynamics, the specific behavior of electrons on a GEB wormhole surface—which interpolates between a catenoid and a cylinder—had not been fully explored. The study aims to determine how the geometric deformation parameter $n$ and the orbital angular momentum $m$ influence the bound states and probability density of the electron.

Methodology
The authors model the system using the spinless stationary Schrödinger equation within the effective mass approximation ( $m^*$ ). The geometry is defined by the GEB wormhole line element, characterized by a throat radius $R$ and a deformation parameter $n$ (where $n$ is an even integer).

Hamiltonian Formulation: The electron is constrained to the surface by a geometric potential. The Hamiltonian includes the kinetic energy term coupled with the induced metric and the da Costa potential ( $V_{dC}$ ), which arises from the mean and Gaussian curvatures of the surface.
Potential Analysis: The authors derive an effective potential ( $V_{eff}$ $V_{e f f}$ ) composed of two competing terms:
- $V_{ind}(u)$ : An induced potential arising from the surface geometry, which is essentially repulsive.
- $V_{dC}(u)$ : The da Costa potential, which is attractive.
  The authors note that the original Hamiltonian contains a non-Hermitian first-order derivative term. However, by applying a variable transformation $\Phi(u) = (u^n + R^n)^{-1/(2n)} y(u)$ , they derive an equivalent Hermitian Schrödinger-like equation with a real spectrum.
Numerical Solution: The resulting differential equation for the effective potential is solved numerically using the finite difference method. The study focuses on specific parameters: a throat radius $R = 70$ Å and an effective electron mass in graphene $m^* = 0.03m_0$ .

Key Contributions and Results
The paper provides a detailed analysis of the effective potential and the resulting bound states for varying values of the deformation parameter $n$ and orbital angular momentum $m$ .

Effective Potential Behavior:
- Case $m=0$ : The attractive da Costa potential dominates. For $n=2$ (conventional catenoid), the potential forms a single well centered at the origin ( $u=0$ ). For $n > 2$ , the geometry creates two points of intense curvature at the ends of the wormhole, transforming the potential into a double well. As $n$ increases (approaching a cylindrical shape), the depth of these wells increases significantly (e.g., from $-13.12$ meV at $n=2$ to $-912.6$ meV at $n=40$ ).
- Case $m \neq 0$ : The centrifugal term modifies the induced potential, making it dominant over the da Costa potential. This generally results in a potential barrier at the origin. However, for sufficiently large $n$ (specifically $n > 16$ for $m=1$ ), the effective potential reverts to a double-well structure, albeit with shallower wells compared to the $m=0$ case.
Bound States and Probability Density:
- $n=2, m=0$ : A single bound state is found with energy $-3.42$ meV. The probability density is a Gaussian function centered at the origin, forming a "probability ring."
- $n > 2, m=0$ : The system exhibits hybrid states due to the proximity of the two potential wells. For $n=10$ , two states are found (ground and first excited). The ground state probability density shows two overlapping Gaussians, while the first excited state shows two distinct rings. As $n$ increases to 40, the wells deepen, and the bound states become more tightly confined.
- $m=1$ : Bound states are only obtained for $n \geq 18$ . Similar to the $m=0$ case, these are hybrid states. For $n=40$ , two bound states are observed with energies $-41.6$ meV and $-30.6$ meV.
Oscillation Frequencies:
The authors calculate the frequency of an electron tunneling between the two wells (oscillating between the ends of the wormhole) using $f = (E_1 - E_0)/h$ .
- For $n=40$ and $R=70$ Å, the frequency is approximately $3.0$ THz for $m=0$ and $2.7$ THz for $m=1$ .
- The study shows that as the throat radius $R$ increases, the potential wells become shallower, and the oscillation frequency decreases (e.g., dropping to $0.3$ THz for $R=200$ Å).

Significance and Claims
The paper claims that the geometry of the GEB wormhole, controlled by the parameter $n$ , fundamentally alters the electronic dynamics on a graphene surface. The transition from a single-well to a double-well potential allows for the existence of hybrid bound states and electron tunneling between the wormhole ends. The authors suggest that the calculated oscillation frequencies (in the Terahertz range) could serve as a characteristic signature of the system.

The work concludes that while the qualitative nature of the potential (single vs. double well) depends on $n$ and $m$ , the quantitative energy levels and confinement are sensitive to the throat radius $R$ . The authors propose that this system offers a theoretical framework for understanding curvature-induced electronic effects in graphene and suggest future extensions involving external fields and thermodynamic properties.

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