How to classifies the distinct "plasma" phases of matter? 
  
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*How to classifies the distinct "plasma" phases of matter?
  

and


  
*What theory classifies the distinct "plasma" phases of matter?
  

According to Wikipedia:
Plasma (from Greek πλάσμα, "anything formed") is one of the four fundamental states of matter, the others being (i) solid, (ii) liquid, and (iii) gas. A (iv) plasma has properties unlike those of the other states.
 For (i) solid, (ii) liquid, and (iii) gas: 

We know that large amount of solid crystals can be classified by the crystal "symmetry (Answer of 1)." And large amount of (i) solid, (ii) liquid, and (iii) gas can be classified by the theory of **Landau-Ginzburg theory of symmetry and symmetry-breaking (Answer of 2).
Some of the solid states have phase transition between the metal phase and the insulator phase, called metal-insulator transition. Some of them can still be classified by the theory of Landau-Ginzburg theory of symmetry and symmetry-breaking,** but some can not. 
 How about the (iv) plasma?



  
*Can plasma be a gapped or gapless phase? Do Landau-Ginzburg theory of symmetry and symmetry-breaking classifies plasma or not? What are the symmetry and the symmetry breaking here? Are the classical E&M plasma and the quantum plasma the different phenomena? Are there something more exotic in plasma other than the Anderson-Higgs mechanism for the plasmon?
  

 A: 
  
*
  
*How to classifies the distinct "plasma" phases of matter?
  
*What theory classifies the distinct "plasma" phases of matter?
  

These questions do not really make sense (unless I am misunderstanding your point).  A plasma is an ionized gas that exhibits a collective behavior similar to a fluid but is a kinetic gas.  There are not multiple phases of plasma.  There are collisional and collisionless plasmas though, i.e., those whose dynamics are governed by particle-particle collisions and those that aren't.
If you are asking what makes a plasma different from three standard phases of matter, then the answer is many things.  For instance, a plasma is governed by instabilities and long-range interactions (i.e., Coulomb collisions).

How about the (iv) plasma?

Plasmas do have extremely large electrical conductivities, but they are not consistent with Type-I superconductors.  In fact, plasmas are very different than both types of solid- and liquid-phase superconductors in that their dynamics are governed by electromagnetic fields.

Can plasma be a gapped or gapless phase? Do Landau-Ginzburg theory of symmetry and symmetry-breaking classifies plasma or not?

I cannot say whether Landau-Ginzburg theory applies to quark-gluon plasmas but it does not apply to electromagnetic plasmas like that in the solar wind.

What are the symmetry and the symmetry breaking here?

I do not think this type of phenomena really matters for most electromagnetic plasmas.  It may be important near pulsars and/or magnetars where spin is very important and the particle gyroradii can be quantized.  However, for most situations a plasma can be treated classically as a kinetic gas.

Are the classical E&M plasma and the quantum plasma the different phenomena?

Yes, electromagnetic plasmas are classical (except for the extreme cases I mentioned before) while quark-gluon plasmas are an entirely different beast mediated by different forces.

Are there something more exotic in plasma other than the Anderson-Higgs mechanism for the plasmon?

I think plasmons are primarily relevant to metals/solids.  I do not think the Anderson-Higgs mechanism is relevant to a classical kinetic gas.
