*Xiao-Gang Wen*

- Published in print:
- 2007
- Published Online:
- February 2010
- ISBN:
- 9780199227259
- eISBN:
- 9780191713019
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199227259.003.07
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Electrons on the interface of two semiconductors can form a new state of matter — fractional quantum Hall (FQH) state — under strong magnetic fields. FQH states cannot be described by Landau symmetry ...
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Electrons on the interface of two semiconductors can form a new state of matter — fractional quantum Hall (FQH) state — under strong magnetic fields. FQH states cannot be described by Landau symmetry breaking theory. These shatter the long-held belief that symmetry breaking theory describes all phases and phase transitions. As a result, a completely new theory is needed to describe FQH states, and this is the topic of this chapter. Many-electron systems in strong magnetic fields and resulting Landau level structures are studied. Laughlin's theory and the hierarchical theory for FQH effect are presented. The chapter then derives the low energy effective Chern–Simons theory for FQH states and discusses the resulting fractional charge and fractional statistics, as well as the K-matrix classification of Abelian FQH states. The theory of chiral gapless edge states is also introduced, where experimental predictions can be made.Less

Electrons on the interface of two semiconductors can form a new state of matter — fractional quantum Hall (FQH) state — under strong magnetic fields. FQH states cannot be described by Landau symmetry breaking theory. These shatter the long-held belief that symmetry breaking theory describes all phases and phase transitions. As a result, a completely new theory is needed to describe FQH states, and this is the topic of this chapter. Many-electron systems in strong magnetic fields and resulting Landau level structures are studied. Laughlin's theory and the hierarchical theory for FQH effect are presented. The chapter then derives the low energy effective Chern–Simons theory for FQH states and discusses the resulting fractional charge and fractional statistics, as well as the K-matrix classification of Abelian FQH states. The theory of chiral gapless edge states is also introduced, where experimental predictions can be made.

*Mark O. Goerbig*

- Published in print:
- 2011
- Published Online:
- September 2011
- ISBN:
- 9780199603657
- eISBN:
- 9780191729515
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199603657.003.0006
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter yields an introduction to quantum Hall effects both for non-relativistic electrons in conventional two-dimensional electron gases (such as in semiconductor heterostructures) and ...
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This chapter yields an introduction to quantum Hall effects both for non-relativistic electrons in conventional two-dimensional electron gases (such as in semiconductor heterostructures) and relativistic electrons in graphene. After a brief historical overview follows a detailed discussion of the kinetic-energy quantisation of non-relativistic and relativistic electrons in a strong magnetic field (section 2). Section 3 is devoted to the transport characteristics of the integer quantum Hall effect, and the basic aspects of the fractional quantum Hall effect are described in section 4. In section 5, several multicomponent quantum Hall systems are briefly discussed, namely the quantum Hall ferromagnetism, bilayer systems and graphene that may be viewed as a four-component system.Less

This chapter yields an introduction to quantum Hall effects both for non-relativistic electrons in conventional two-dimensional electron gases (such as in semiconductor heterostructures) and relativistic electrons in graphene. After a brief historical overview follows a detailed discussion of the kinetic-energy quantisation of non-relativistic and relativistic electrons in a strong magnetic field (section 2). Section 3 is devoted to the transport characteristics of the integer quantum Hall effect, and the basic aspects of the fractional quantum Hall effect are described in section 4. In section 5, several multicomponent quantum Hall systems are briefly discussed, namely the quantum Hall ferromagnetism, bilayer systems and graphene that may be viewed as a four-component system.

*Xiao-Gang Wen*

- Published in print:
- 2007
- Published Online:
- February 2010
- ISBN:
- 9780199227259
- eISBN:
- 9780191713019
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199227259.003.08
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

According to the principle of emergence, the properties of material are mainly determined by how the atoms are organized in the material. Such organization is formally called order. The vast range of ...
More

According to the principle of emergence, the properties of material are mainly determined by how the atoms are organized in the material. Such organization is formally called order. The vast range of materials is a result of the rich variety of orders that atoms can have. For a long time, it has been believed that all orders are described as symmetry breaking. A comprehensive theory for phases and phase transitions is developed here based on the symmetry breaking picture. The existence of FQH states (and superconducting states) indicates that there are new states of matter that cannot be described as symmetry breaking. Completely new theory is needed to describe those new states of matter. This chapter outlines the theory of topological order and theories of quantum order for the new states of matter, such as FQH states. Many new concepts and new language, such as topology-dependent degeneracy, fractional statistics, edge states, etc, are introduced to describe new states of matter.Less

According to the principle of emergence, the properties of material are mainly determined by how the atoms are organized in the material. Such organization is formally called order. The vast range of materials is a result of the rich variety of orders that atoms can have. For a long time, it has been believed that all orders are described as symmetry breaking. A comprehensive theory for phases and phase transitions is developed here based on the symmetry breaking picture. The existence of FQH states (and superconducting states) indicates that there are new states of matter that cannot be described as symmetry breaking. Completely new theory is needed to describe those new states of matter. This chapter outlines the theory of topological order and theories of quantum order for the new states of matter, such as FQH states. Many new concepts and new language, such as topology-dependent degeneracy, fractional statistics, edge states, etc, are introduced to describe new states of matter.