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The discovery of novel topological phase advances our knowledge of nature andstimulates the development of applications. In non-Hermitian topologicalsystems, the topology of band touching exceptional points is very important.Here we propose a real-energy topological gapless phase arising fromexceptional points in one dimension, which has identical topological invariantsas the topological gapless phase arising from degeneracy points. We develop agraphic approach to characterize the topological phases, where the eigenstatesof energy bands are mapped to the graphs on a torus. The topologies ofdifferent phases are visualized and distinguishable; and the topologicalgapless edge state with amplification appropriate for topological lasing existsin the nontrivial phase. These results are elucidated through a non-HermitianSu-Schrieffer-Heeger ladder. Our findings open new way for identifying topologyphase of matter from visualizing the eigenstates.

Visualizing Topology of Real-Energy Gapless Phase Arising from Exceptional Point