Contrary to classical phase transitions, quantum phase transitions can only be accessed by varying a physical parameter - such as magnetic field or pressure - at absolute zero temperature.

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From Wikipedia

The nature of Phase transition is the competition between order (some constraints, such as pressure, external magnetic field) and disorder (such as temperature)

What place in this classification take Charge Density Wave transition?

It depends on the materials you are talking about. If you are thinking of He gas and liquid and solid then phase transition between them is certainly a quantum phenomenon, especially liquid-superfluid transition. If you wish to understand microscopically then you must invoke quantum mechanics. However gas-liquid and liquid-solid and solid-solid transitions have universal behavior and these can be understood from classical statistical mechanics to a great extent. Now if you are thinking of for example H2O then most of the things you may be able to understand classically. However to understand quantitative behavior you may again have to invoke quantum statistics. Zero-point vibrational amplitude gives you information about the quantum fluid and this amplitude is large for quantum fluids like He and also to some extent H but heavier liquids like O, N, H2O should behave like classical fluids.

@Jatendra: You have worked on the superfluid He. So you must be knowing better than us.

The second order quantum phase transition between two ground states of a many body quantum system is a transiton at T=0 due to a competition between two ground states due to interactions represented by a PHYSICAL PARAMETER P of the Hamiltonian at P=P_crit . When changing P , there occurs an ordered phase at PP_crit . In a statistical system at T nonzero, one has to account for the averaging over the respective statistical distributions. The result of the competition between the phases is temperature-dependent and the phase transition occurs at T=T_crit. and is called temperature or classical PT. That means that either the order supporting mechanism (interaction) or a disorder supporting mechanism prevails according to the temperature.. Usually increase of the temperature supresses the phase transition which exists at T=0.

The transition to CDW phase in the electron ground state of the interacting electron system or due to its interaction with phonons in low dimensions D=1,2 is the transition in the electron system which exists both at T=0 as a quantum PT and can survive at T nonzero as a classical (themodynamic) phase transition.

@ Tapan. Honestly, if you can believe, I never considered myself an expert of any subject because once you do that you contradict the basic instinct of research. I simply consider myself as a student of the subject and wish to know the subject more deeply. It was this spirit with which I asked this question at this forum and it was this spirit when I joined research gate. If you see different posts you would find that we have different understanding of the related subject and we need to come as close as possible which we can do only by intense discussion. For an example your statement, " Now if you are thinking of for example H2O then most of the things you may be able to understand classically. However to understand quantitative behavior you may again have to invoke quantum statistics." seems to indicate that what we can understand as a classical change needs quantum statistics for its understanding at quantitative scale. Since quantitative understanding is always the aim of physics, quantum nature seems dominate even the transformation of water vapours to liquid water and transformation of liquid water to solid water. During the course of my thoughts I also got this perception but I could not conclude to myself that this could be acceptable to physicists at large. However, your post has helped me to make my question to be more specific, viz. whether vapour to liquid transition and liquid to ice transition of $H_2O$ is a classical transition or a quantum transition; in this context it is important to note that energy spectrum of water molecule follows quantum laws not the classical ones.

@Jatendra: You are right. The appropriate mechanics for the atom and molecules is of course the quantum mechanics. The appropriate theory of a large assembly of atoms and molecules is the quantum statatistics. But classical mechanics and statistical mechanics can however give some understanding of the assembly of atoms and molecules but one can get often wrong results. So one must always be careful while using them.

Sure we all have only incomplete understanding and therefore it is necessary to discuss with others. The opportunity for that is unique in Research Gate.

@Eva Let me thank you for your post which tries address the difference between quantum PT and classical PT. However, could you pl. elaborate on your two statements for their better understanding.

"That means that either the order supporting mechanism (interaction) or a disorder supporting mechanism prevails according to the temperature.. Usually increase of the temperature supresses the phase transition which exists at T=0."

This is a little bit confusing because as per your statement (above the quoted statement) increasing pressure at T = 0 leads to a state of disorder from order and increasing T at constant pressure also leads to a state of disorder from that of order. If I understand right increase in pressure increases interaction and the phase at P > P$_{crit}$ as per your statement is disordered phase. In view of this how interaction is order supporting mechanism ?

"The transition to CDW phase in the electron ground state of the interacting electron system or due to its interaction with phonons in low dimensions D=1,2 is the transition in the electron system which exists both at T=0 as a quantum PT and can survive at T nonzero as a classical (themodynamic) phase transition."

The part of this statement which is not clear is "can survive at T nonzero as a classical (themodynamic) phase transition."

How a transition at T=0$^+$ (just above 0 K) becomes classical. Note that it is the heart of my question and answer to this would help me understand which transition is classical and which is quantum.

Depends on what you're after. My guess is that the notion of phases and phase transitions existed long before quantum mechanics.

Yatendra, probably a misunderstanding occured: P denotes here a parameter of Hamiltonian, supporting the ordered state, e.g. strength of electron-phonon interaction, not pressure.