**2013.12.13**

Jun ONOE Research Laboratory for Nuclear Reactors, Associate Professor jonoe[at]nr.titech.ac.jp Lab. HP |

## Publication

J. Onoe, T. Ito, H. Shima, H. Yoshioka and S. Kimura: “Observation of Riemannian geometrical effects on electronic states”, *Europhysics Letters* **98**, 27001 (2012).

DOI: 10.1209/0295-5075/98/27001

## Abstract

In 1916, Einstein first applied Riemannian geometry to explain the distortion of time-space by a gravitational field, by introducing the metric tensor used in Riemannian geometry. Four years later, his prediction was confirmed by the observation of a gravitational lens (see Fig. 1). This is the first example demonstrating that light propagates in a curved space.

Fig. 1. Gravitational lens (geometric curvature) effects in light. We usually cannot see the galaxy (or star) just behind the planet. But, if the planet is very heavy, the time-space near the planet is distorted by its gravity, which makes the direction of light emitted from the galaxy (or star) changed. Thus we can see the galaxy (or star) even just behind the planet.

What happen to electrons when they move in a curved space? Since the application of Riemannian geometry to quantum mechanics in 1950s, it is curious to clarify whether or not the geometric curvature term consisting of the average and Gaussian curvatures appeared in the lateral Hamiltonian for electron motion in a periodic uneven curved surface affects electronic properties. However, because no materials enabling us to examine the curvature effects have been synthesized so far, nobody has confirmed the effects on electrons until now.

Our laboratory has hitherto found one-dimensional (1D) metallic uneven peanut-shaped C_{60} polymer (see the up-left side of Fig. 2) formed from electron-beam irradiation of a pristine C_{60} film in an ultrahigh vacuum chamber. We recently examined in situ high-resolution photoemission spectra of the 1D polymer, and obtained the result in good agreement with theoretical prediction by our group [*Phys. Rev. B* **79**, 201401(R) (2009)]. This is the first observation demonstrating that the electron motion is affected by the geometric curvature term due to the 1D uneven curved surface (see the right-down side of Fig. 2).

Fig. 2. Schematic illustration of the 1D metallic uneven peanut-shaped C_{60} polymer (left-up side) and of electron motion along the 1D uneven curved surface (right-down side).

More recently, it is found that modern mathematics such as Riemannian prediction, topology, and eight- number, etc. is strongly related to natural science, which opens a door bringing us to a new paradigm.

Based on the present work, we aim to establish a new academic system fusing materials science with modern geometry for the first time in the world, by elucidating mathematical correlation between physical and geometric quantities.

## References

**Related original papers**

- J. Onoe, T. Nakayama, M. Aono, and T. Hara: "Structural and electrical properties of an electron-beam irradiated C
_{60}film",*Appl. Phys. Lett.***82**, 595–597 (2003). - J. Onoe, A. Nakao, and A. Hida: "Valence photoelectron spectra of an electron-beam irradiated C
_{60}film",*Appl. Phys. Lett.***85**, 2741–2743 (2004). - J. Onoe, T. Itoh, S. Kimura, K. Ohno, Y. Noguchi, and S. Ueda: “Valence electronic structure of cross-linked C
_{60}polymers:*In situ*high-resolution photoelectron spectroscopic and density-functional studies”,*Phys. Rev. B***75**, 233410 (2007). - Y. Toda, S. Ryuzaki, and J. Onoe: “Femtosecond carrier dynamics in an electron-beam-irradiated C
_{60}film”,*Appl. Phys. Lett.***92**, 094102 (2008). - H. Shima, H. Yoshioka, and J. Onoe: “Geometry-driven shift in the Tomonaga-Luttinger exponent of deformed cylinders”,
*Phys. Rev. B***79**, 201401(R) (2009). - J. Onoe, A. Takashima and Y. Toda: “Infrared phonon anomaly of one-dimensional metallic peanut-shaped C
_{60}polymer”,*Appl. Phys. Lett.***97**, 241911 (2010). - J. Onoe, A. Takashima, S. Ono, H. Shima, and T. Nishii: “Anomalous enhancement in the infrared phonon intensity of one-dimensional uneven peanut-shaped C
_{60}polymer”,*J. Phys.*:*Condens. Matter***24**, 175405 (2012).

**Review papers**

**J. Onoe**：”One-dimensional metallic peanut-shaped nanocarbon with positive and negative Gaussian curvatures: Toward a new science of quantum electronic systems on Riemannian surface”,*J. Surf. Sci. Soc. Jpn.***30**, 659–666 (2009).**J. Onoe**, T. Ito, S. Kimura, H. Shima, Y. Toda, and H. Yoshioka: “One-dimensional uneven peanut-shaped C60 polymer: a quantum electronic system on Riemannian space” (invited paper),*Fullerene, Nanotubes, and Carbon Nanostructures***20**, 1-16 (2012).

## Keywords

One-dimensional metallic peanut-shaped C_{60} polymer, geometric curvature, Tomonaga-Luttinger liquid, photoemission spectroscopy, Riemannian geometry