Matter can show a wide variety of states and forms, including liquids, solids and gases, and metals, semiconductors, insulators and superconductors. Research in the past few years has revealed a new class of materials known as topological insulators and topological superconductors, which have attracted much attention in the field of condensed matter physics. A key feature of topological insulators/superconductors is their ‘massless’ particle behavior. Akira Furusaki, chief scientist in the Condensed Matter Theory Laboratory at the RIKEN Advanced Science Institute (ASI), and colleagues have created a classification table to explain the kinds of topological insulators/superconductors that can exist theoretically, leading to the discovery that Majorana particles, first predicted more than 70 years ago, are present in certain topological superconductors.

New states of matter

The discipline of geometry dealing with properties that remain constant even under continuous deformation is known as topology. In this field, all objects that can be transformed from one to another by continuous deformation are considered identical. For example, continuously deforming a coffee cup can produce a toroidal form, and conversely, a toroid can be deformed into the shape of a coffee cup (Fig. 1). Coffee cups and toroids are therefore regarded as being ‘topologically identical’. However, to transform a coffee cup into a figure with two openings requires the additional step of making another opening, and so cannot be reached by simple continuous deformation of the coffee cup. Coffee cups are therefore ‘topologically different’ from a figure-of-eight shape.

“Topology has not been widely employed in physics to date,” says Furusaki. “However, research over the past few years has shown that, unlike in known insulators and superconductors, electron states—or ‘wavefunctions’—in certain substances that are now called topological insulators and topological superconductors possess topological ‘numbers’. In quantum physics, an assembly of electron states in a substance can be mathematically regarded as a ‘space’. The number characterizing the shape of the space is the topological number.”

“The first topological insulator was actually discovered in 1980, although it wasn’t recognized as such at the time. It was a material exhibiting an integer quantum Hall effect.” The classical Hall effect is the production of an electric potential at right angles to an injected current and an applied magnetic field in conducting materials. The integer quantum Hall effect is an analogous phenomenon in quantum physics, usually manifesting itself only on the scale of atoms and electrons. At ultralow temperatures, however, certain quantum phenomena, such as the quantum Hall effect, can manifest at the macroscopic level. Superconductivity, the disappearance of electrical resistivity at low temperature, is another example of low-temperature quantum-based phenomena.

“To observe the integer quantum Hall effect, we apply an intense magnetic field perpendicular to the plane of an electron system confined to a two-dimensional planar interface between two semiconductors at ultralow temperature. The ratio of the resulting electric current to the voltage produced at right angles with respect to the current and magnetic field is an integer multiple of a number made of fundamental constants in quantum physics. This is a manifestation of quantum physics on a macroscopic scale. Any electron system in this state is a topological insulator, with an integer value for the topological number.”

The importance of this discovery of the integer quantum Hall effect is reflected by the fact that its discoverer received the Nobel Prize in Physics in 1985, just five years after the groundbreaking discovery. “However, it was another major discovery—the quantum spin Hall effect—that has led to recent advances in research into topological insulators.”

Spin refers to a property possessed by electrons and other particles, analogous to clockwise and counterclockwise rotations and referred to as up-spin and down-spin. “These spins are involved in spin–orbit interaction, by which certain two-dimensional substances exhibit the integer quantum Hall effect on ‘spin flows’, even without application of a magnetic field. American researchers predicted in 2005 that this state would be a ‘Z2 topological insulator’ with a binary topological number (0 and 1). This is a manifestation of the quantum spin Hall effect.”

This phenomenon was verified experimentally in 2007, and the suggestion that the Z2 topological insulator also occurs in three-dimensional substances was verified experimentally in 2008.

“Subsequent research has shown theoretically that an insulator exhibiting an integer quantum Hall effect or quantum spin Hall effect can be understood to represent a novel state of matter called a topological insulator, and that certain superconductors can also become topological superconductors characterized by topological numbers.”

Massless particles

“One fascinating feature of topological insulators/superconductors is their ‘massless’ particle behavior,” says Furusaki. Massless particles move around freely at the edges of a two-dimensional topological insulator/superconductor, or over the entire surface of a three-dimensional topological insulator/superconductor.

“A topological insulator and the insulating vacuum surrounding it have different topological numbers. Massless particles emerge at the interfaces of regions of differing topological numbers, such as at the edges and surfaces of a substance. At the edges of a two-dimensional material exhibiting the quantum spin Hall effect, for example, up-spin and down-spin electrons are moving in mutually opposite directions (Fig. 2). Although electrons have a mass equivalent to 1/1,836th of a proton, the up-spin and down-spin electrons moving at the margins behave as massless particles, moving according to the Dirac equation of motion for electrons in a vacuum but assuming a zero electron mass, and so can be deemed massless particles.”

The presence of electrons that behave as massless particles was first verified experimentally in 2005 in graphene, a substance distinct from topological insulators/superconductors. Graphene is carbon in the form of a flat, honeycomb lattice just a single atom thick. The electrons in graphene behave as massless particles, moving at a speed close to 1/300th of the speed of light. This is several times faster than the speed of electrons in silicon, such as in modern electronics. For this reason, graphene is also expected to serve as a material for ultra-high-speed computers in the future.

In 2009, senior research scientist Naoya Tajima and others in the ASI’s Condensed Molecular Materials Laboratory led by Chief Scientist Reizo Kato verified the presence of electrons behaving as massless particles in the organic semiconductor known as α-(BEDT-TTF)2I3, attracting a great deal of attention. “Massless particles are an exciting subject of research for condensed matter physicists such as myself,” says Furusaki.

Classifying topological insulators/superconductors

“The research into topological insulators/superconductors that I have described has principally been conducted by researchers in the US. We worked together with collaborators in the USA to clarify the kinds of topological insulators and superconductors that can exist theoretically.

“Electrons move freely around the surfaces of topological insulators and superconductors. In an ordinary material, an impurity or defect in the crystal lattice causes the electrons to become localized and immobilized at that site due to interference between electron waves, resulting in what is known as an Anderson localization, where electric currents no longer flow,” says Furusaki (Fig. 3). “In a topological insulator or superconductor, however, the surface allows the electrons to move on freely, even if they encounter impurities. We applied the theory of Anderson localization, and determined the conditions under which electrons would flow freely without being localized. Figuring out when these conditions can be met, we made our classification table.”

Discovery of particles predicted by a genius more than 70 years ago

The classification of these conditions has led to some interesting findings. “We know that when helium-3 is cooled to an ultralow temperature, a state known as ‘superfluid B phase’, or 3He-B, emerges. This was found to be classifiable as a ‘topological superfluid’ under category DIII.”

When cooled to a temperature near absolute zero, 3He liquefies, and becomes a superfluid with further cooling. As a superfluid, all viscosity is lost; when superfluid helium is placed in a cylindrical vessel and caused to flow in some way, the flow continues without slowing. As such, superfluidity is another manifestation of quantum physics at the macroscopic level.

“On the surface of superfluid 3He-B, which is a topological superfluid, excited 3He atoms move around freely as massless particles. To our surprise, we demonstrated that these particles theoretically represent Majorana particles.”

The existence of Majorana particles was predicted by the Italian physicist Ettore Majorana more than 70 years ago. “Majorana was a physicist and genius,” explains Furusaki, “but he became a legend when he went missing the year after he made the prediction, when he was still in his early 30s.”

In 1929, before Majorana went missing, the British physicist Paul Dirac predicted the existence of ‘antiparticles’ with a charge and other properties opposite from the ordinary particles that form a material. “For example, he deduced theoretically that negatively charged electrons should have a counterpart, positively charged positrons, and that positively charged protons should be partnered by negatively charged antiprotons.” In 1932, positrons were discovered in cosmic rays, confirming the presence of antiparticles.

“Majorana predicted in 1937 the existence of particles that are electrically neutral and do not permit any distinction between particles and antiparticles, that is, particles that are their own antiparticle. Those are now called Majorana particles. Although the existence of such particles has not been confirmed experimentally, the neutrino, an elementary particle, has been hypothesized to be one kind of Majorana particle.”

These suggestions by Furusaki and others have made Majorana particles the focus of research in the field of topological insulators/superconductors and superfluids. “Majorana particles should exist not only in the superfluid B phase of 3He, but also at the boundaries of certain topological superconductors. We recently published a paper proposing an experimental method for demonstrating the existence of Majorana particles.”

At the ASI, the Low Temperature Physics Laboratory headed by Chief Scientist Kimitoshi Kono ranks foremost in the world in physical experiments of low-temperature phenomena such as 3He superfluidity. “I want to cooperate with Kono and others in contributing to the experiments aimed at confirming the existence of Majorana particles.”

Other research groups are proposing applications of Majorana particles to quantum computers, a concept for a futuristic computer capable of rapidly calculating solutions to problems that would take thousands of years to resolve with currently available supercomputers due to the vast numbers of computations required. However, in developing the quantum computer, the ‘coherent’ state for quantum physical phenomena quickly collapses under outside influences, posing a major problem. According to Furusaki, “Majorana particles formed on the surfaces or at the edges of topological superconductors are predicted to be unlikely to be affected by any external influence and should therefore be capable of retaining their coherence for a long time. For this reason, researchers are considering ways to apply Majorana particles to quantum computers.”

Exploring the properties of matter

Before starting research on topological insulators and superconductors, Furusaki’s primary research theme was strongly correlated electron systems that achieve high-temperature superconductivity and the like. “Although strongly correlated electron systems remain one of my major research themes, for several years now I have been focusing my efforts on the study of topological insulators and superconductors. Any number of unimagined and fascinating phenomena may be awaiting discovery here. In several years time, I could be engaged in investigating unknown quantum phenomena in materials quite distinct from topological insulators and superconductors—after all, curiosity is central to being a researcher.” The buds of the truly innovative sciences and technologies needed to support the societies of the future will no doubt sprout from basic research based on such unconfined curiosity.

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About the Researcher

Akira Furusaki

Akira Furusaki was born in Hatogaya in Saitama, Japan, in 1966. Graduating from the Faculty of Science at The University of Tokyo in 1988, he obtained his PhD in physics in 1993 from the same university. After becoming a research associate in the Department of Applied Physics at The University of Tokyo in 1991, he worked as a postdoctoral associate from 1993 to 1995 in the Department of Physics at the Massachusetts Institute of Technology, USA. Soon after returning to Japan, he was appointed as an associate professor at the Yukawa Institute for Theoretical Physics, Kyoto University in 1996. Since October 2002, he has been chief scientist in the Condensed Matter Theory Laboratory at RIKEN. His research focuses on the search for new states of matter and the development of quantum theory for electronic transport, superconductivity and magnetism.