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Introduction to Topological Order

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Quantum Information Meets Quantum Matter

Part of the book series: Quantum Science and Technology ((QST))

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Abstract

In primary school, we were told that there are four states of matter: solid, liquid, gas, and plasma. In college, we learned that there are much more than four states of matter. For example, there are ferromagnetic states as revealed by the phenomenon of magnetization and superfluid states as defined by the phenomenon of zero-viscosity. The various phases in our colorful world are extremely rich. So it is amazing that they can be understood systematically by the symmetry-breaking theory of Landau. However, in the past 20–30 years, we discovered that there are even more interesting phases that are beyond Landau symmetry-breaking theory. In this chapter, we discuss new “topological” phenomena, such as topological degeneracy, that reveal the existence of those new phases—topologically ordered phases. Just like zero-viscosity defines the superfluid order, the new “topological” phenomena define the topological order at macroscopic level.

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Notes

  1. 1.

    Note that real-life superconductivity can be described by the Ginzburg–Landau theory with a dynamical U(1) gauge field. The condensation of charge 2e electron pair breaks the U(1) gauge theory into a \(\mathbb {Z}_2\) gauge theory at low energies. A \(\mathbb {Z}_2\) gauge theory is an effective theory of \(\mathbb {Z}_2\) topological order. Thus, a real-life superconductor has a \(\mathbb {Z}_2\) topological order. In many textbooks, superconductivity is described by the Ginzburg–Landau theory without the dynamical U(1) gauge field, which fails to describe the real-life superconductors with dynamical electromagnetic interaction. Such a textbook superconductivity is described by a U(1) symmetry breaking.

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Authors and Affiliations

  1. University of Guelph, Guelph, ON, Canada

    Bei Zeng

  2. Physics, California Institute of Technology, Pasadena, CA, USA

    Xie Chen

  3. Institute of Physics, Chinese Academy of Sciences, Beijing, China

    Duan-Lu Zhou

  4. Massachusetts Institute of Technology, Cambridge, MA, USA

    Xiao-Gang Wen

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  1. Bei Zeng
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Correspondence to Bei Zeng .

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Zeng, B., Chen, X., Zhou, DL., Wen, XG. (2019). Introduction to Topological Order. In: Quantum Information Meets Quantum Matter. Quantum Science and Technology. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-9084-9_6

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  • DOI: https://doi.org/10.1007/978-1-4939-9084-9_6

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