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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 thanfour states of matter. For example, there are ferromagnetic states as revealedby the phenomenon of magnetization and superfluid states as defined by thephenomenon of zero-viscosity. The various phases in our colorful world are sorich that it is amazing that they can be understood systematically by thesymmetry breaking theory of Landau. In this paper, we will review the progressin last 20 -- 30 years, during which we discovered that there are even moreinteresting phases that are beyond Landau symmetry breaking theory. We discussnew "topological" phenomena, such as topological degeneracy, that reveal theexistence of those new phases - topologically ordered phases. Just likezero-viscosity defines the superfluid order, the new "topological" phenomenadefine the topological order at macroscopic level. As a new type of order,topological order requires a new mathematical frame work, such as fusioncategory and group cohomology, to describe it. More recently, we find that, atmicroscopical level, topological order is due to long-range quantumentanglements, just like fermion superfluid is due to fermion-paircondensation. Long-range quantum entanglements lead to many amazing emergentphenomena, such as fractional quantum numbers, fractional/non-Abelianstatistics, and perfect conducting boundary channels. Long-range quantumentanglements can even provide a unified origin of light and electrons (or moregenerally, gauge interactions and Fermi statistics): light waves (gauge fields)are fluctuations of long-range entanglements, and electrons (fermions) aredefects of long-range entanglements.

Topological order: from long-range entangled quantum matter to an unification of light and electrons