New chapters of condensed matter physics——topological quantum phases of matter beyond the Landau—Ginzburg—Wilson paradigm

In the history of condensed matter physics, the Landau―Ginzburg continuous phase transition theory provided the understanding of the condensed ordered phases and their related phase transitions. Combined with the Wilson renormalization group theory, the Landau―Ginzburg―Wilson paradigm finally became the cornerstone of modern physics. However, since the discoveries of the quantum Hall effect, fractional quantum Hall effect, and higher temperature copper-oxide superconductors, there emerged more and more exotic quantum phases of matter beyond the Landau―Ginzburg―Wilson paradigm, opening new chapters of condensed matter physics. In this article, we briefly review the basic physics of several important topological quantum phases based on the integer quantum Hall effect of twodimensional electron systems. Then we focus on intrinsic topologically ordered phases of matter in two-dimensional strongly correlated electron systems. In particular, for the toric code model proposed by Kitaev for quantum computation, we discuss its topological ordered wave functions, ground-state degeneracy, topological anyonic excitations, and the possible topological quantum phase transitions. In addition, we outline some recent important theoretical progresses and possible future directions.