Can spontaneous symmetry breaking order develop in a topological phase?
Landau's symmetry breaking theory is the basis of modern condensed matter physics, from which most phases can be understood. The recent discovery of topological insulators simulates the study on the topological phase which is beyond this theory. An interesting question is whether the topological phase and spontaneous symmetry breaking order can coexist. We study this problem based on the topological phase in the Haldane model. Firstly by including a staggered potential, we show that if the system is subjected to the symmetry needed by the charge density wave (CDW) order in advance, the topological phase can coexist with the CDW order, realizing the topological CDW phase. While when the nearest neighor interaction which preserves the symmetry of the origin Hamiltonian is included, the CDW order can not develop from spontaneous symmetry breaking in the topological phase. The results imply that a spontaneous symmetry breaking order may not develop in a topological phase.