There are several proposed types of exotic matter: Negative mass would possess some strange properties, such as accelerating in the direction opposite of applied force.
Despite being inconsistent with the expected behavior of "normal" matter, negative mass is mathematically consistent and introduces no violation of conservation of momentum or energy.
The closest known real representative of such exotic matter is the region of pseudo-negative-pressure density produced by the Casimir effect.
Because time travel is considered to be non-physical, tachyons are believed by physicists either not to exist, or else to be incapable of interacting with normal matter.
For instance, Na3Cl is a two-dimensional metal, made of layers of pure sodium and salt that can conduct electricity.