Choreographed entanglement dances: Topological states of quantum matter

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Science  22 Feb 2019:
Vol. 363, Issue 6429, eaal3099
DOI: 10.1126/science.aal3099

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A topological paradigm shift

The discovery of topological phases of matter forced condensed matter physicists to question and reexamine some of the basic notions of their discipline. Wen reviews the progress of the field that took a sharp turn from Landau's broken symmetry paradigm to arrive at the modern notions of topological order and quantum entanglement in many-body systems. This development was made possible by using increasingly sophisticated mathematical formalisms.

Science, this issue p. eaal3099

Structured Abstract


Our world is very rich. One aspect of its richness is reflected in the existence of many different phases of matter. More than half a century ago, Landau developed a theory to describe phases of matter on the basis of symmetry breaking. He pointed out that the distinction between different phases stems from the way their constituent particles are organized (ordered); different phases correspond to different symmetries of the particles’ ordering. For many years, it was widely believed that the symmetry-breaking theory described all phases and all phase transitions.


However, the study of chiral spin liquids and quantum Hall (QH) liquids eventually revealed phases of matter and organizations of particles not described by the symmetry-breaking theory. This new kind of order was referred to as topological order, because it is closely related to the topological quantum field theory introduced by Witten in 1989.

It took researchers 20 years to realize that topological order is nothing but the patterns of quantum entanglement in many-body systems, which can be intuitively understood via two analogies to dancing: (i) Dance of particles (or step dance). Particles move in a spiral fashion and take a fixed number of steps to dance around each other. (ii) Dance of strings. The local degrees of freedom form strings that join in a particular way (see the figure). The strings can dance by moving around and reconnecting freely.

The first type of dance, the step dance, describes topological order in chiral spin liquids and QH liquids, whereas the string dance describes topological order in other spin liquids. The QH liquids have been realized by electron systems at the interface of semiconductors and by graphene, under strong magnetic fields. The topological order categorized by the string dance may be realized by electron spins in certain materials, such as herbertsmithite and RuCl3.

In the string-net liquids described by the string dance, the strings can be viewed as the “electric” flux of a gauge theory, and the string density wave give rise to an emergent (non-Abelian) gauge field. The ends of the strings are topological excitations that may carry fractional charges, fractional (non-Abelian) statistics in two-dimensional (2D) systems, and Fermi statistics in 3D systems. The QH liquids categorized by the step dance also have emergent gauge theory—the Chern-Simons gauge theory. This type of dance leads to indestructible perfect conducting boundaries, as well as indestructible qubits (units of quantum information). Topologically ordered states are materials with intriguing properties, which may be useful in electronic devices and topological quantum computation.


The emergence of topological phases of matter from the patterns of many-body quantum entanglement is a truly new phenomenon. New mathematics is needed to describe and classify topological orders. Recent studies have revealed that a unitary modular tensor category is required to classify 2D bosonic topological orders, and unitary braided fusion categories are necessary to classify 2D fermionic topological orders. To classify 2D topological orders with symmetry G, a G-cross unitary modular tensor category (for bosons) or a unitary braided fusion categories over Rep(G) (for bosons and fermions) is needed. However, the mathematical theory, including higher-category theory, to classify topological orders in three dimensions and beyond is still evolving.

Many-body entanglement is not only the origin of many new states of quantum matter (such as topological orders), it is also the origin of emergent gauge fields, as well as emergent Fermi or fractional statistics, from the simple bosonic qubits that form the system. Recent work has indicated that our empty space itself might be a system formed by many qubits—a qubit ocean. In other words, the space itself is formed by entangled qubits; if there is no qubit, there is no sense of space. The entanglement of the qubits provides a sense of neighborhood and dimension of the space. If the quantum entanglement of the qubits in the ocean is described by a particular string-net dance, then the string density waves in the string-net liquid generate electromagnetic waves that satisfy the Maxwell equation and gluon waves that satisfy the Yang-Mills equation. String ends produce electrons and quarks that carry Fermi statistics and satisfy the Dirac equation. Those emergent gauge fields and fermions are the elementary particles in the standard model. Such an emergence picture based on string-nets represents a unification of matter and information (see the figure).


Local degrees of freedom form strings (colored lines), which can then join in a specific way to create a string-net. Sting-net liquids can give rise to emergent (non-Abelian) gauge theory, emergent fractional or non-Abelian statistics (in 2+1D), and emergent fermions (in 3+1D). This concept unifies gauge interaction and Fermi statistics and provides a cohesive origin for light and electrons, as well as other elementary particles.


It has long been thought that all different phases of matter arise from symmetry breaking. Without symmetry breaking, there would be no pattern, and matter would be featureless. However, it is now clear that for quantum matter at zero temperature, even symmetric disordered liquids can have features, giving rise to topological phases of quantum matter. Some of the topological phases are highly entangled (that is, have topological order), whereas others are weakly entangled (that is, have symmetry-protected trivial order). This Review provides a brief summary of these zero-temperature states of matter and their emergent properties, as well as their importance in unifying some of the most basic concepts in nature.

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