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About the BookThis excellent text covers a year's course in advanced theoretical electromagnetism, first introducing theory, then its application. Today only, save on used books and help us reach 15,000 books sold to double the donations. Figure 1: The classical Hall effect—electrons (moving blue spheres) flowing in an external magnetic field (between ‘S’ and ‘N’) experience a force at right angles to both the field and the current flow.
When a conductor carrying electrical current is placed in a magnetic field, the flowing electrons experience a force perpendicular to both the magnetic field and the current. At very low temperatures, the classical Hall effect breaks up and displays some striking quantum characteristics. Now, Akira Furusaki at the RIKEN Advanced Science Institute in Wako and co-workers in Japan and the USA have shown that the quantum Hall effect is strongly affected by boundaries at the edge of a material1. During the classical Hall effect, the electrons moving under the influence of the Lorentz force experience resistance to their flow.
The quantum version of the Hall effect was discovered in 1980 when researchers measured the properties of electrons confined to just two dimensions at very low temperatures near absolute zero. The size of each step is determined by two fundamental constants, the electron charge e and Planck’s constant h, regardless of the material being studied.
Many theories have been proposed to explain exactly how electrons move between the resistance plateaus in the quantum Hall effect. At such low temperatures the electrons experience a phenomenon called Anderson localization—they are scattered so much that they cannot propagate over any distance, and effectively stay in one place. The main theoretical challenge is to work out what happens to delocalize the wavefunctions, allowing the system to jump between adjacent plateaus.
Previous research completed by Furusaki and co-workers has helped them to explain the Anderson localization. They found that the electron distribution obeys so-called multifractal statistics, meaning that they follow similar patterns on both small and large scales.

The researchers realized that the boundary differences will influence the quantum Hall effect.
The quantum Hall effect depends on impurities or defects in the sample, which can be thought of as hills that the electrons must climb over or skirt around. Furusaki and colleagues recalculated these critical wavefunctions for electrons undergoing a transition between plateaus near the edge of a sample.
Any new theories for the quantum Hall effect will have to take these constraints into account. 1980 Revised Edition, corrected unabridged version of 1972 edition, clean book, looks new except for light foxing of edges. Topics include vectors D and H inside matter, conservation laws for energy, momentum, invariance, form invariance, covariance in special relativity and more.
The production of voltage, or potential difference, by this (Lorentz) force is known as the Hall effect (Fig. In fact, this so-called integer quantum Hall effect reveals quantizations that are so precise they have been used to accurately determine some important numbers in quantum mechanics, including the fine-structure constant that characterizes the strength of electromagnetic forces. Here, the Hall resistance looks very different—it jumps up in quantized steps as the magnetic field strength increases, producing a series of plateaus (Fig. This quantum Hall effect is so precisely quantized that it is now used as a standard of resistance measurement. This means that their wavefunctions are very narrow, and the sample effectively acts as an insulator against the Hall effect. In one study2, they examined the electron wavefunctions in materials that can undergo transitions from metallic (electrically conducting) to insulating behavior. However, the electrons at the sample’s boundary edges showed quite different distributions from those in the bulk of the sample.
Previous calculations have missed this subtlety by using bulk physical quantities that are valid only in the center of a sample.

At the sample edges, there are limits to the directions the electrons can travel to overcome these obstacles, so their dynamics are different. Instead, there are ‘edge wavefunctions’ extended along the edge of the sample which can carry current,” explains Furusaki. They found that the transitions do not follow the same multifractal statistics that have been assumed in previous studies.
Furusaki looks forward to unraveling the final details of this remarkable example of quantum physics in action. In a different study3, Furusaki and his collaborators have already explained an unusual quantum Hall effect caused by the relativistic nature of electrons in graphene, which could eventually replace silicon in integrated circuits. He graduated from Faculty of Science, the University of Tokyo in 1988 and obtained his PhD in physics in 1993 from the same university. However, it turns out that these restrictions at the edges are crucial in producing the quantum Hall effect. Boundary multifractality at the integer quantum Hall plateau transition: Implications for the critical theory. He became a research associate at Department of Applied Physics, the University of Tokyo in 1991, and worked as a postdoctoral associate for two years at Department of Physics, Massachusetts Institute of Technology in USA, before being appointed as an associate professor at Yukawa Institute for Theoretical Physics, Kyoto University in 1996.
It is exploited in several electronic devices such as flow sensors, joysticks, car ignitions and even spacecraft propulsion.
Since October 2002 he has been a chief scientist of Condensed Matter Theory Laboratory at RIKEN. Multifractality and conformal invariance at 2D metal-insulator transition in the spin-orbit symmetry class.

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