Symmetry-protected topological invariants of symmetry-protected topological phases of interacting bosons and fermions

Recently, it was realized that quantum states of matter can be classified as long-range entangled (LRE) states (i.e. with non-trivial topological order) and short-range entangled (SRE) states (\ie with trivial topological order). We can use group cohomology class ${\cal H}^d(SG,R/Z)$ to systematically describe the SRE states with a symmetry $SG$ [referred as symmetry-protected trivial (SPT) or symmetry-protected topological (SPT) states] in $d$-dimensional space-time. In this paper, we study the physical properties of those SPT states, such as the fractionalization of the quantum numbers of the global symmetry on some designed point defects, and the appearance of fractionalized SPT states on some designed defect lines/membranes. Those physical properties are SPT invariants of the SPT states which allow us to experimentally or numerically detect those SPT states, i.e. to measure the elements in ${\cal H}^d(G, R/Z)$ that label different SPT states. For example, 2+1D bosonic SPT states with $Z_n$ symmetry are classified by a $Z_n$ integer $m \in {\cal H}^3(Z_n, R/Z)=Z_n$. We find that $n$ identical monodromy defects, in a $Z_n$ SPT state labeled by $m$, carry a total $Z_n$-charge $2m$ (which is not a multiple of $n$ in general).Comment: 42 pages, 12 figures, 3 tables, RevTeX4-

Zoo of quantum-topological phases of matter

What are topological phases of matter? First, they are phases of matter at zero temperature. Second, they have a non-zero energy gap for the excitations above the ground state. Third, they are disordered liquids that seem have no feature. But those disordered liquids actually can have rich patterns of many-body entanglement representing new kinds of order. This paper will give a simple introduction and a brief survey of topological phases of matter. We will first discuss topological phases that have topological order (ie with long range entanglement). Then we will cover topological phases that have no topological order (ie with only short-range entanglement).Comment: 18 pages, 8 figures, 4 tables. A short review, expanded versio

From new states of matter to a unification of light and electrons

For a long time, people believe that all possible states of matter are described by Landau symmetry-breaking theory. Recently we find that string-net condensation provide a mechanism to produce states of matter beyond the symmetry-breaking description. The collective excitations of the string-net condensed states turn out to be our old friends, photons and electrons (and other gauge bosons and fermions). This suggests that our vacuum is a string-net condensed state. Light and electrons in our vacuum have a unified origin -- string-net condensation.Comment: 14 pages, to appear in YKIS2004 proceedings, homepage http://dao.mit.edu/~we

A theory of 2+1D bosonic topological orders

In primary school, we were told that there are four phases of matter: solid, liquid, gas, and plasma. In college, we learned that there are much more than four phases of matter, such as hundreds of crystal phases, liquid crystal phases, ferromagnet, anti-ferromagnet, superfluid, etc. Those phases of matter are so rich, it is amazing that they can be understood systematically by the symmetry breaking theory of Landau. However, there are even more interesting phases of matter that are beyond Landau symmetry breaking theory. In this paper, we review new "topological" phenomena, such as topological degeneracy, that reveal the existence of those new zero-temperature phases -- topologically ordered phases. Microscopically, topologically orders are originated from the patterns of long-range entanglement in the ground states. As a truly new type of order and a truly new kind of phenomena, topological order and long-range entanglement require a new language and a new mathematical framework, such as unitary fusion category and modular tensor category to describe them. In this paper, we will describe a simple mathematical framework based on measurable quantities of topological orders $(S,T,c)$ proposed around 1989. The framework allows us to systematically describe/classify 2+1D topological orders (ie topological orders in local bosonic/spin/qubit systems)..Comment: 35 pages, 20 figures, 12 tables. RevTeX