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arXiv:cond-mat/0406441 · Annals of Physics

An introduction to quantum order, string-net condensation, and emergence of light and fermions

Xiao-Gang Wen

Published Mar 1, 2005• 42 citations
arXivHTMLPDF
DOI
10.1016/j.aop.2004.07.001

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Recently, it was pointed out that quantum orders and the associated projective symmetry groups can produce and protect massless gauge bosons and massless fermions in local bosonic models. In this paper, we demonstrate that a state with such kinds of quantum order can be viewed as a condensed state of nets of strings. The emergent gauge bosons and fermions in local bosonic models can be regarded as a direct consequence of string-net condensation. The gauge bosons are fluctuations of large closed string nets. The ends of open strings are the charged particles of the corresponding gauge field. For certain types of strings, the ends of open strings can even be fermions. According to the string-net picture, fermions always carry gauge charges. This suggests the existence of a new discrete gauge field that couples to neutrinos and neutrons. We also discuss how chiral symmetry that protects massless Dirac fermions can emerge from the projective symmetry of quantum order.

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Recently, it was pointed out that quantum orders and the associated projective symmetry groups can produce and protect massless gauge bosons and massless fermions in local bosonic models. In this paper, we demonstrate that a state with such kinds of quantum order can be viewed as a condensed state of nets of strings. The emergent gauge bosons and fermions in local bosonic models can be regarded as a direct consequence of string-net condensation. The gauge bosons are fluctuations of large closed string nets. The ends of open strings are the charged particles of the corresponding gauge field. For certain types of strings, the ends of open strings can even be fermions. According to the string-net picture, fermions always carry gauge charges. This suggests the existence of a new discrete gauge field that couples to neutrinos and neutrons. We also discuss how chiral symmetry that protects massless Dirac fermions can emerge from the projective symmetry of quantum order.
This book has three types of theories: boson, fermion, and gauge theories. It asks which theories describe the elementary particles in our universe. Surprisingly, elementary particles are described by the two more complicated and non-local theories: fermion theory and gauge theory. In the last chapter and in this chapter it is demonstrated that fermion and gauge theory are not really needed. Boson theory may be able to explain everything, since both fermion theory and gauge theory can emerge from local bosonic models (or qbit models), if bosons (qbits) are organized to have a string-net condensed order. This chapter studies soluble quantum spin models to explain string-net condensation. It finds that gauge bosons are simply collective excitations associated with string density wave (with the electric field corresponding to the string density). The fermons can appear as the ends of open strings. Thus the string-net order explains the emergence of gauge bosons and fermions. It unifies gauge interaction and Fermi statistics. The chapter shows that non-orientable strings give rise to Z_2 gauge theory while the orientable string give rise to U(1) gauge theory.
For most of the last century, condensed matter physics has been dominated by band theory and Landau's symmetry breaking theory. In the last twenty years, however, there has been an emergence of a new paradigm associated with fractionalization, emergent gauge bosons and fermions, topological order, string-net condensation, and long range entanglements. These new physical concepts are so fundamental that they may even influence our understanding of the origin of light and electrons in the universe. This book is a pedagogical and systematic introduction to the new concepts and quantum field theoretical methods in condensed matter physics. It discusses many basic notions in theoretical physics which underlie physical phenomena in nature, including a notion that unifies light and electrons. Topics covered include dissipative quantum systems, boson condensation, symmetry breaking and gapless excitations, phase transitions, Fermi liquids, spin density wave states, Fermi and fractional statistics, quantum Hall effects, topological/quantum order, and spin liquid and string-net condensation. Methods discussed include the path integral, Green's functions, mean-field theory, effective theory, renormalization group, bosonization in one- and higher dimensions, non-linear sigma-model, quantum gauge theory, dualities, projective construction, and exactly soluble models beyond one-dimension.