  Since the well-known PT symmetry has its fundamental significance and
implication in physics, where PT denotes the combined operation of
space-inversion P and time-reversal T, it is extremely important and intriguing
to completely classify exotic PT-invariant topological metals and to physically
realize them. Here we, for the first time, establish a rigorous classification
of topological metals that are protected by the PT symmetry using KO-theory. As
a physically realistic example, a PT-invariant nodal loop (NL) model in a 3D
Brillouin zone is constructed, whose topological stability is revealed through
its PT-symmetry-protected nontrivial Z2 topological charge. Based on these
exact results, we propose an experimental scheme to realize and to detect
tunable PT-invariant topological NL states with ultracold atoms in an optical
lattice, in which atoms with two hyperfine spin states are loaded in a
spin-dependent 3D OL and two pairs of Raman lasers are used to create
out-of-plane spin-flip hopping with site-dependent phase. Such a realistic
cold-atom setup can yield topological NL states, having a tunable ring-shaped
band-touching line with the two-fold degeneracy in the bulk spectrum and
non-trivial surface states. The states are actually protected by the combined
PT symmetry even in the absence of both P and T symmetries, and are
characterized by a Z2-type invariant (a quantized Berry phase). Remarkably, we
demonstrate with numerical simulations that (i) the characteristic NL can be
detected by measuring the atomic transfer fractions in a Bloch-Zener
oscillation; (ii) the topological invariant may be measured based on the
time-of-flight imaging; and (iii) the surface states may be probed through
Bragg spectroscopy. The present proposal for realizing topological NL states in
cold atom systems may provide a unique experimental platform for exploring
exotic PT-invariant topological physics.
