How come metal isn't considered a state of matter? I know in chemistry metals are a class of elements on the periodic table, but in physics metal is more like a state of matter. All of the elements that are called metals on the periodic table are metals under normal conditions.
Apart from the 4 classical states of matter, Wikipedia lists a lot more, not only extreme ones, like quark-gluon plasma, BEC, superfluids, supersolids, but also liquid crystals, states that just have different magnetic properties and even glass is considered as a unique state. Metal isn't on the list, nor is it included as a state of matter in any other literature I've managed to find.
The reason I'm confused about this is because elements that are nonmetallic actually have metallic states at high pressures, including hydrogen, helium, oxygen, carbon, other heavier nonmetals. Even water, a compound, has a metallic state at 4000K, 100GPa. Metals have nonmetallic allotropes too.
Also, the primary difference between a metal and a nonmetal is analogous to the difference between a gas and plasma, which are different states of matter.
Another state of matter was discovered not long ago called the Jahn-Teller Metal. If this is classified as a state of matter, how come metal can't be classified as one too?
 A: The problem is just that there are two different meanings of "state of matter"- the commonplace meaning, and the meaning used by physicists.
The commonplace meaning is that a state of matter is either solid, liquid, or gas. For some reason, plasma is now also often included. The definitions given usually involve rigidity and whether the substance fills its container.
The physicist meaning of state of matter is more like any substance with a given set of properties*, and so it has many more possibilities. For example, certain solids can change between being magnetic and being non-magnetic at a critical temperature, called the Curie temperature. Even though both are solids, they are still considered different states of matter in the physicist sense. Furthermore, in some cases the transition from liquid to gas is really a smooth crossover without any sharp transition, and in this case the two aren't really considered to be separate phases in the physicist sense. So the definitions are really quite different- one isn't just a generalization of the other. Unfortunately, popular descriptions almost always mix the two up, especially in a list like the Wikipedia list of phases.
So, then, is a metal a phase of matter? In the commonplace sense, it is not- the possible phases are solid, liquid, and gas, and metals can be either solid or liquid. However, in the physicist sense metals are indeed a phase, or depending on the system there may be multiple metallic phases. The study of metal-insulator transitions of various kinds is a major focus of condensed matter physics. The defining characteristic of a metal has varied somewhat over time and depending on context, but it is often either (1) A decreasing electric resistivity with decreasing temperature, (2) Extended (as opposed to localized) single-electron wavefunctions, or (3) A gapless (as opposed to gapped) energy spectrum.
*the formal definition generally involves some parameter that exists only within one of the phases, called an "order parameter"
A: It all really depends on how you define a state of matter, and there is no clear definition at all. For example, if you look on Wikipedia you will find this "definition"

In physics, a state of matter is one of the distinct forms in which matter can exist.

which gives almost no information (what exactly is meant by "form" here, and how do we distinguish them?).
However (as also reported in the Wikipedia article), there is another, more precise way to define a state of matter, which is the one that I personally prefer.
Indeed, while defining a state of matter can be complicated, defining a phase transition is relatively easier, since phase transition are characterized by well-defined particular behaviors (divergences, discontinuities, power-law behaviors with specific critical exponents...) of certain thermodynamic quantities (1). With this definition, we can say that some material went from a state of matter to another when it underwent a phase transition.
If we adopt this definition, then in some cases we can consider the "metallic state" as a state of matter. A recent work (2) has for example pointed out that the insulator-metal transition in hydrogen has the characteristics of a first order phase transition.
Therefore, it really seems like in some cases you could think of a metal as a distinct state of matter.

(1) I am including both first and second order phase transition in this very generic definition.
(2) Evidence of a first-order phase transition to metallic hydrogen, Mohamed Zaghoo, Ashkan Salamat, and Isaac F. Silvera Phys. Rev. B 93, 155128 (2016).
A: My thinking is that states of matter are characterized by the behavior of the atoms or other basic constituents of a substance. In contrast, whether the substance is metallic (or insulating, or semiconducting, or superconducting) is an electronic property, classifying the behavior of the electrons. Naturally, the electronic properties are highly dependent on the configuration and behavior of the atoms. That's why, e.g., some allotropes of carbon are metallic (metallic nanotubes), while others are insulating (diamond), and still others are zero-bandgap semiconductors (various graphene thicknesses). Their atoms are arranged in different ways (and exist in differing numbers of spatial dimensions: 1, 2, 3!), leading to different electronic properties. But they are all solids because their atoms remain fixed in place as they do in solids. 
If you look at your list of different states of matter, only the magnetic states can be considered a description of the electronic properties of a substance rather than of the atoms--the exception that proves the rule! For after all, electrons themselves can be modeled as existing as one of the states of matter (ever heard of Fermi liquid theory or the hydrodynamic theories of conduction?). Metals aren't another form of matter, the electrons in them are just funny liquids!
Anyway, don't get too caught up in these sorts of classifications. There's no governing body of classifying states of matter. The dirty little secret of this sort of thing is that we're all just making it up as we go along! 
