The answers to your above question are manifold. Nevertheless, I will try to answer your query qualitatively.

Fluctuation means disturbance or perturbation or symmetry breaking. Phase transition is also one kind of fluctuation e.g. transition of solid to liquid or liquid to gas. (the formation of bubbles exactly at the boiling point). But even at absolute zero temperature, where all motion in the classical world is frozen out, special quantum mechanical fluctuations prevail that can drive the transition between two quantum phases. You can think of zero point energy of a simple harmonic oscillator (E = ħw). So this can be taken as an example of quantum fluctuation.

Now, let us come to dimensionality. The quantum fluctuation increases as one moves from 3D to 1D. Let us consider the case of 3D 1st. In 3D, the particles are free to move in any directions, which merely perturbs the system. Now if we need to squeeze the system to low dimension, then we need to reduce the size of the system i.e. volume and size of the particle simultaneously. With due process, we will end up at a position where, we cannot exceed x.px > ħ, such that the particle tries to disturb the system severely. At this point, we say there is quantum fluctuation.

And regarding the last part of your question, quantum phase transitions (QPT) occur at zero temperature as a function of pressure, field, chemical composition etc. and are driven by quantum fluctuations rather than thermal fluctuations. The vicinity of QPT is quantum critical points, which controls the non-zero temperature behavior. Please follow the attached article to know further regarding QPTs, especially in magnetism.

Link - Article Quantum phase transitions

Thanks Prashanta Mukherjee for a very nice and lucid explanation. I will go through the link.

Dear Prof. Subodha Sahoo

I am trying to relate your question to 1D Peierls transition [1,2], also to the physical understanding of a quantum phase transition. Not results although. Looks like I don´t understand what does it mean a quantum phase transition.

See for instance:

[1] pp 432 of L. Landau and E. Lifshitz, Vol. 6, Statistical Physics, Pergamon 1980, Part I.

[2] https://en.wikipedia.org/wiki/Peierls_transition

Dear Pedro L. Contreras E. ,

Peierls transition has a analogous feature in case of quantum magnets (especially quantum AFMs) and is known as spin-Peierls transition. Below such transition a uniform array of magnetic ions (uniform spin chains) get distorted and form alternating configuration such that the exchange interaction between a pair of magnetic atoms is no longer uniform . This occurs mainly due to lattice instability. Due to the bond alternation, two interactions alternate along the chain direction. This drives the quantum magnets to a gapped state. Such a gap can be closed by application external magnetic field. When the gap is closed, the system moves to a long-range-ordered state and a quantum phase transition occurs. (From a gapped to gapless state)

You can follow the link for more information: Phys. Rev. B 14, 3036 (1976).

Thank you so much for the instructive explanation & reference Dear Prof. Prashanta Kumar Mukharjee

I understand now that these kind of Pierls transitions may occur because of a lattice instability (phonon or spin lattice). So I guess quantum phase transitions aren´t related to lattice instability at all, are they? I would highly appreciate your comment about this misundertsanding of mine.

Dear Pedro L. Contreras E. ,

Yes you are absolutely correct Peierls transition does not falls under a quantum phase transition, since it is driven by temperature. Whereas, quantum phase transition is driven by non-thermal parameters. (e.g. magnetic field, pressure, chemical composition etc.)

The transition from a gapped to gappless state is driven by magnetic field is a quantum phase transition.

Thank you again for the clarification Dear Prof. Prashanta Kumar Mukharjee

I belong to the old solid state theorist school, before the invention of the quantum phase transitions-QPT. Now, I understand that QPT transitions are a subset of the whole family of magnetic transitions & as you stated clearly "...from gapped to gappless state driven by a magnetic field..."

Dear @ Prashanta Kumar Mukharjee

There are several papers on quantum phase transition where researchers has probed a transition as a quantum phase transition even though quantum fluctuation is not dominant around that temperature. So my question lies there how to really comment on a phase transition as a quantum phase transition at finite temperature?

Dear Biswajit Dutta,

QPT without any quantum fluctuations seems to be improbable. Maybe the average order parameter verges to zero but the individual fluctuations will never be. Also in experiments, we set a temperature window, which ranges from *sufficiently low* to 'upper critical range'. Thus, one has to really stick to the small window in order to achieve QCP. Anyway please share any article claiming QPT w/o any quantum fluctuations.

Dear Prashanta Kumar Mukharjee

I didn't try to say QPT with out quantum fluctuation , I just wanted to ask at any finite temperature (lets say 8K or 10K) what criteria will decide the phase transition will be a QPT? and what do u mean by individual fluctuation?

please look at this link Article Quantum critical point and intermediate valence fluctuations...

Dear Biswajit Dutta ,

So, the fluctuation is any kind of disturbance. In a fictitious way, let's say we both are fighting with as much energy as we can put. But, at some point one, either mine or your energy will overcome and one will win. Like that, in magnetism, there is are competitions. When one competes with another, and ordering occurs.

I am not experienced in itinerant systems. Anyway, I will try to address your query in view of the paper suggested by you. So, the first thing is we are probing groundstate properties and we need to lower our temperature so that their thermal fluctuations will die completely. The paper which you have provided says that the ground state relies on the competition between RKKY and Kondo effects. At some particular doping conc. the ground state changes from RKKY to Kondo. The crossover between these two states is driven by the chemical disorder. And this is QPT since non-Fermi-Liquid behaviour has been observed. Please read the first paper attached by me in this question's thread. But now you ask, if I get some transition by simply doping then is that transition belong to QPT??? The first answer is no. One has really look for other parameters, like how heat capacity is behaving at lower temperatures. Usually, at low temp. Heat capacity is linear according to Fermi-liquid theory, but if there is deviation then yes there is something else and one needs further understanding to know its QPT or not.

Dear Prashanta Kumar Mukharjee

Thanks for your valuable suggestion, actually I am totally new in this field and facing several questions while going in to more details. I will study further and get back to u.

Dear Biswajit Dutta ,

You are welcome. I am also getting new questions while going deep which is a good sign. Let's discuss more in future.

Dears Biswajit Dutta and Prashanta Kumar Mukharjee, I have had the same question in my mind for a while:

How to comment on a quantum phase transition at finite temperature?

Are that QFT at finite T, what was called many years ago as Lifshitz 2½ topological electronic phase transitions?.

Lifshits transition of 2 1/2 type. The original paper is here: JETP, Vol. 11, No. 5, p. 1130 (November 1960)

http://www.jetp.ac.ru/cgi-bin/dn/e_011_05_1130.pdf

Just in case, I have a thread on that possessing a related question, although I didn't mention the word QFT.

https://www.researchgate.net/post/Do_we_understand---geometry_changes---involved_in_a_212_Lifshitz_electronic_Fermi_surface_phase_transition

I will make in a future post, some comments about the fluctuations.