  We propose topological semimetals generated by the square-root operation for
tight-binding models in two and three dimensions, which we call square-root
topological semimetals. The square-root topological semimetals host topological
band touching at finite energies, whose topological protection is inherited
from the squared Hamiltonian. Such a topological character is also reflected in
emergence of boundary modes with finite energies. Specifically, focusing on
topological properties of squared Hamiltonian in class AIII, we reveal that a
decorated honeycomb (decorated diamond) model hosts finite-energy Dirac cones
(nodal lines). We also propose a realization of a square-root topological
semimetal in a spring-mass model, where robustness of finite-energy Dirac
points against the change of tension is elucidated.
