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We investigate emergent topological gapless phases in the square-latticeKitaev model with additional hopping terms. In the presence of nearest-neighborhopping only, the model is known to exhibit gapless phases with two topologicalgapless points. When the strength of the newly added next-nearest-neighborhopping is smaller than a certain value, qualitatively the same phase diagrampersists. We find that further increase of the extra hopping results in a newtopological phase with four gapless points. We construct a phase diagram toclarify the regions of emergent topological gapless phases as well astopologically trivial ones in the space of the chemical potential and thenext-nearest-neighbor hopping strength. We examine the evolution of the gaplessphases in the energy dispersions of the bulk as the chemical potential varies.The topological properties of the gapless phases are characterized by thewinding numbers of the present gapless points. We also consider the ribbongeometry to examine the corresponding topological edge states. It is revealedthat Majorana-fermion edge modes exist as flat bands in topological gaplessphases. We also perform the analytical calculation as to Majorana-fermionzero-energy modes and discuss its implications on the numerical results.

Emergence of new topological gapless phases in the modified square-lattice Kitaev model