*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 ...
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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.

*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/9780199227259.003.06
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

The last few chapters have discussed many-boson theory and many-fermion theory because nature has many-boson systems and many-fermion systems. Do many-boson/many-fermion theories describe all the ...
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The last few chapters have discussed many-boson theory and many-fermion theory because nature has many-boson systems and many-fermion systems. Do many-boson/many-fermion theories describe all the systems in nature? The answer is no. Nature also has photons. Although photons are bosons, they cannot be described by the local bosonic theory discussed in Chapter 3. Photons are described by a non-local quantum theory which is called U(1) gauge theory. This chapter discusses U(1) gauge theory as well as Z_2 gauge theory on lattice. It stresses the non-localness in their Hilbert space. The gapless photonic, as well as the ‘electric’ and the ‘magnetic’ excitations are studied. The chapter also discusses the confinement phase transition in those gauge theories, and a duality between U(1) gauge theory and XY model in 1+2 dimensions.Less

The last few chapters have discussed many-boson theory and many-fermion theory because nature has many-boson systems and many-fermion systems. Do many-boson/many-fermion theories describe all the systems in nature? The answer is no. Nature also has photons. Although photons are bosons, they cannot be described by the local bosonic theory discussed in Chapter 3. Photons are described by a *non-local* quantum theory which is called U(1) gauge theory. This chapter discusses U(1) gauge theory as well as Z_2 gauge theory on lattice. It stresses the non-localness in their Hilbert space. The gapless photonic, as well as the ‘electric’ and the ‘magnetic’ excitations are studied. The chapter also discusses the confinement phase transition in those gauge theories, and a duality between U(1) gauge theory and XY model in 1+2 dimensions.