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Topological Aspects in Solids State Physics

June 03, 2008 – June 22, 2008


Tokyo, Japan


Mahito Kohmoto

Masaki Oshikawa

The integer quantum Hall effect is a remarkable quantum phenomenon induced by a magnetic field which breaks time reversal symmetry. Quantized Hall conductance here is related to a topological invariant, which is called as Chern number or TKNN integer. For example, a band insulating state with vanishing Chern number cannot be connected smoothly to a quantum Hall state with a finite Chern number. This means that the quantum Hall states are characterized topologically.

During the last quarter century after the discovery of the deep relation between the integer quantum Hall effect and the topology, condensed matter theory based on topological concepts has been developing gradually but steadily. For example, in the Laughlin state which exhibits the fractional quantum Hall effect, the quasiparticles obey anyon statistics rather than Bose statistics or Fermi statistics. This fact originates from topological properties of the wave functions.

Recently, topological concepts are of increasing importance more than ever. It has been proposed that spin currents can be controlled by an electric field. This “spin Hall effect” is a hot topic with potential industrial applications. Although the spin Hall effect differs much from the quantum Hall effect as physical phenomena, its theory is based on the topological theory of the quantum Hall effect.

Moreover, the recent experimental reports of the integer quantum Hall effect in single-layer graphene renewed interest in the topological structure of electrons in two-dimensional systems. Many studies are based on Dirac fermions in 2+1 dimensions which involves the continuum approximation and describes the low-energy modes in graphene. Thorough understandings, however, would require general topological descriptions. Use of the exact solution of the fundamental model might be prospective.

Quantum spin systems have been studied for a long time as a typical problem in quantum many-body problems. Spontaneous breaking of symmetries and the corresponding order parameters are basic concepts for classifications of various phases. The novel phases and the phase transitions outside this framework are attracting much attention, recently. A useful and seemingly fundamental concept in describing them is “topological order” which has arisen from the study of the fractional quantum Hall effect.

In this way, many of the recent important topics in solid state physics are rather closely related to topological concepts. At this workshop, we aim to gain unified understandings of various novel phenomena from topological perspective.

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