So according to this, scientists have discovered more than 5 states of matter we usually had that is the solid, liquid, gases, and Bose-Einstein-Condensate, and plasma. So how many topological phases of matter are discovered is it one, ten, or hundreds
The "states of matter" classification is a useful way to understand matter in broad strokes, but it has limits to its usefulness. In particular, not all phase transitions correspond to changes in the "state of matter" occupied in the broad-strokes classification.
As one simple example, magnetic materials experience a phase transition (known as the "Curie point") when you heat them up: if you start with a ferromagnet with a permanent magnetization and you heat it up, there is a sharp point where it will lose that permanent magnetization. From the point of view of thermodynamics, this phase transition shows the same behaviour as the melting of a solid into a liquid. But would you count "permanent magnet" and "demagnetized magnet" as different "states of matter"? Or are they both just "solid"?
Similarly, there are phase transitions within each "state of matter", particularly within solids. One good example is water ice, which can crystalize into about twenty different crystal structures, depending on the temperature and pressure. And, if you change the temperature and pressure so that you cross the boundary into a different crystal structure, you get a phase transition with the same thermodynamics again.
To make things worse, even the clear separation lines in our paradigm can get blurred. In normal conditions, liquids and gases are distinct and clearly recognizable, and you have a thermodynamic phase transition (evaporation / condensation) when you go from one to the other. But, if the temperature is high enough, you can get something called a supercritical fluid which has properties of both, and which can connect smoothly to both a liquid and a gas when you cool it down, depending on the pressure, without ever passing through a phase transition. So, should liquids and gases be considered as just one single "state of matter"?
OK, so, that said $-$ what about topological phases? This is a hard question to answer, because each topological phase is (in some ways) specific to the individual system where it appears, while also (in other ways) very easily identifiable across a range of different systems. There is a rigorous classification of the different topological invariants that characterize topological phases, colourfully referred to as the periodic table of topological invariants, and this gives a good sense of what is possible in this area: probably around one or two dozen, or several hundred, depending on whether you count each phase as corresponding to one individual invariant or to a full specification of several. Again, this isn't an area where any hard-and-fast statements can really pull any weight, and there's a heck of a lot more physics to discover and understand before we can be sure that we've found everything, but that "periodic table" is a good picture of our current understanding.
This is a very interesting question, and one that I am pleased to see a high school student asking.
The first thing to realize is that this is a difficult question to answer even if we restrict ourselves to non-topological ("trivial") phases of matter. The standard high school picture that there are three phase of matter (or four or five, depending on who you ask) is woefully incomplete. As Emilio Pisanty points out in his answer, there are something like twenty distinct phases of liquid water distinguished by their crystalline structure—and that's just water! Other materials may have other crystalline structures and associated solid phases. There are also solids that do not have a well-defined crystal structure, like glasses1, and there are phases that are intermediate between liquids and crystalline solids. So even if we just focus on the simple solid/liquid/gas distinction, there are very many more phases than we are used to thinking about.
However, the solid/liquid/gas picture is not the end of the story. The distinction between these kinds of phases, from a microscopic point of view2, is in how the atoms or molecules that make them up are arranged in space. We can also distinguish phases of matter based on other features, like their electric or magnetic properties, as Emilio also mentions. A piece of matter that is intrinsically magnetic (like a bar magnet) is in a different phase from a piece of matter that does not spontaneously produce its on magnetic field. Considering all these different ways that matter can order itself enlarges our classification further.
With such a zoo of phases of matter, it might seem hopeless to try to count them all. Indeed, if by "count them" we mean literally writing down the number of different phases, I'm afraid we're out of luck. However, it turns out there is a very elegant way of classifying or labelling different trivial phases based on their symmetry. Since you're in high school, I won't unnecessarily complicate things by trying to explain in mathematical detail what I mean by symmetry or exactly what how this classification works. A simple cartoon example suffices: a solid in which the particles organize themselves into a hexagonal pattern is different from one in which the particles form a cubic pattern, and both of these are different from liquids, in which there is no apparent pattern. This symmetry classification is based on mathematical structures called groups, and, in principle, it gives us a mathematically well-defined way of saying what all the different trivial phases are.
All of this is before we even begin to contemplate topological phases. Indeed, prior to the discovery of topological phases like the quantum Hall fluids, physicists believed they had completed the classification of phases of matter—that it was all just symmetry. The experimental observation of integer quantum Hall phases showed that there can be phases of matter that have all the same symmetries, but are nevertheless distinct from each other. Studies of the quantum Hall phases, as well as another theoretically proposed class of phases called quantum spin liquids, led to the development of the notion of topological order. Topological orders are topological phases of matter without symmetries, and microscopically, different topological orders correspond to different patterns of quantum entanglement between microscopic particles.
While the problem of classifying all of the different topological orders is not completely solved, there is by now quite a well-developed theory of how to classify topological orders, as well as much active work at extending this theory3. Like the symmetry classification of trivial phases is based on the mathematics of groups, the classification of topological orders is based on the mathematics of categories4. As is the case with the symmetry classification of trivial orders, it seems unlikely we will be ever be able to say that "there are exactly this many topological orders," and write down a number, but it seems likely we will have a mathematically well-defined way of saying what all the different possible topological orders are.
Topological orders are topological phases without symmetry, and our zoo of phases becomes richer still if we include topological phases with symmetry. These include symmetry-protected topological (SPT) and symmetry-enriched topological (SET) orders. SPT orders are distinct from topological orders in that they are no longer topological if we remove the symmetry. SET phases are like topological orders in that they are intrinsically topological; if we remove the symmetry of an SET phases, we are left with a topological order. As with topological orders, the classification of SPT and SET orders is ongoing. One approach to classifying SPT orders is based on the mathematics of cohomology5. For the case of systems of non-interacting fermions with certain kinds of symmetry (called topological insulators and topological superconductors), this classification is embodied in the "periodic table" that Emilio mentioned in his answer. There is also a theory of SPT and SET phases based on category theory6.
Finally, I point out that there is still another class of phases of matter beyond the aforementioned trivial and topological phases. These are the so-called fracton or non-liquid phases7. These phases challenge our understanding of how matter can organize (and ultimately, therefore, or understanding of quantum systems) as much if not more than topological phases have.
So how many different topological phases are there? The answer is that we don't know—and we probably never will in the concrete sense you might hope for—but that new tools are currently in development to be able to "count" them as much as we possibly can8. Since the question of what phases of matter there are is ultimately a question of what kind of quantum systems there can be, and since all systems are quantum systems, this means that exciting new work is currently in progress to determine what the fundamental laws allow nature to look like.
- Since this is complicated enough already, let's not dwell on whether glasses count as solids or not.
- This is not the only point of view, or even the most useful one. Usually we are most interested in the macroscopic or thermodynamic properties of a piece of matter.
- For those interested in the details, this paper and its references are good places to start.
- Part of the reason that the classification of topological orders is incomplete is because the purely mathematical theory of categories is still under development, and indeed new tools in category theory have been developed in the effort in classify topological orders.
- See https://arxiv.org/abs/2001.01722 for a review, and https://arxiv.org/abs/2002.02433, https://arxiv.org/abs/2002.05166, https://arxiv.org/abs/2002.12932 for detailed constructions.
- Unfortunately, it is clear based on the classification so far that the answer is much larger than 42, so we will have to discover a new question to which that number is the answer.