Let us have one homogeneous and isotropic gas, that is the highest symmetry. If you condense it other bodies then it is clear that lost symmetries. In magnetism happens that when you go below a critical temperature there is a breaking of symmetry arisen goldstone particles (the spins align in a given direction below the critical temperature). Thus is seems that the highest state with symmetries is one without structure and very hot.

   One general form to analyse these broken symmetries mathematically is to introduce a Landau Hamiltonian where an order parameter can tell us its behaviour under a thermal change.

   I hope that this can help.

The general goldstone mode derivation , you are referring to , presumably in an (electro-)magnetic background field, is not always valid. This depends on what symmetry you think about in microscopic terms is broken, and whether this symmetry has ,  w o u l   b e  long range forces or genuine long range forces associated with it.

     Dear Peter,

     In magnetism, which was the field mentioned in the question, the paramagnetic phase has the symmetry given by SO(3), because the magnetic moments are free to rotate in three dimensions. But when you reach a ferromagnetic phase, for instance, SO(3) breaks spontaneously this symmetry to SO(2) because one direction is fixed. In this case the associated Goldstone are the magnons,i.e. spin waves whose local magnetization oscillates. 

Dear Daniel,

Yes , you give the complete and correct  answer in the case of magnetism .

Dear Daniel,

  Thanks for your direct and comprehensive explaination for the spontaneous magnetization's effect.It impressed me.But I'm still puzzled by the 'scale' problem.What I want to emphasize is that how to use scale factor describing the symmetry degree of different scales system.For instance , we can adjust cutoff  'Lambda' (equal to setting up smallest space interval )to make theory effective under different scale,such as four-fermion interaction theory which work on the energy(available energy) smaller than 80.4GeV(invariant mass of W+,- boson).This method(what I know) was used to develope renormalization theory,accurately,renormalization group.Futhermore, I want similar method of quantifying scale and make qualitative analysis first to provide a picture.

   Thanks for your attention.

    The name of renormalization group is a little confused in the literature. It was born in QED and in fact is not at all a group. The invariance of a physical system under a scale transformation is followed by the conformal group which is very well known in mathematics.

But, unfortunately, this is not the main confusion. In QED identities as the one of Ward-Takahashi plays a fundamental role helping to prove that at certain infinitessimal cut-off the charge and mass of an electron is hidden and this helps to avoid the classical singularities.

    To get such aim in condensed matter is made in very different form. Long time ago L.P.Kadanoff introduced the trick of using "spin-blocks" using the Ising model of magnetism to implement the phase transition para-ferro changing temperature close to the Curie temperature. The idea is that the correlation functions between spins are infinite at the critical temperature. K.G. Wilson could implement these concepts and new technical methods to solve problems as the Kondo magnetism too. In 1970's this subject was fashion and many interesting papers where written.

  A good review is:

    arXiv:1310.5533

Dear Professor Daniel,

  Yes, we cannot adjust cutoff 'Lambda' in renormalization,but setup the threory with a finite cutoff.And system would present corresponding properties at different energy process. It's my fault.Thanks for your correction on my false understanding.

  But,you may misunderstand my mention of renormalization group.I didn't say it could be a group.A group should based on 4 condition: Closure , Associativity , Identity element , Inverse element.I have read "A Critical History of Renormalization" written by Professor Kerson Huang.Another book he wrote "Fundamental Forces of Nature:The Story of Gauge Fields " also clearly presented a comprehensive explaination of RG,RG trajectory and UV(and IR) fixed pointed. Those concepts gave a picture of how renormalized parameters(running couplings) changed in different effective theory(QED--IR fixed point,QCD--UV fixed point)at all length scales .

  Finally,I'm glad of finding your answer gave me a picture that how correlation functions between spins act at the critical temperature. There might be some more intrinsic relation between symmetry principle and correlation function(furthemore fundamental laws).It needs in-deep analysis.

   Dear Tianyou,

   You know that the idea of correlation functions and its use in calculations is mainly done in quantum field theory.  I recommend you two references that can help you to progress in understanding better it.

   S:K.Ma, Modern Theory of Critica Phenomena, Frontiers in Physics, 1976.

   N.Goldenfeld, Lectures of Phase Transitions and the Renormalization Group, Frontiers in Physics, 1992.

Thank you professor.

    Dear Remi,

    What is not correct? If you want to speak about exchange interaction, magnetic domains or domain walls(Neel or Bloch ones) that is correct, but never in contradiction with the spontaneous breaking of symmetry at the critical temperature. In the case of ferromagnetism this is Curie temperature or Neel temperature for antiferromagnetism, when the correlaction function goes to the infinity.

   In the ferromagnetic phase there are more energies and couplings that the quantum exchange interaction J between spins, although it is the fundamental one. By the way the magnetic dipoles (classical coupling) are not fundamental in ferromagnetism, only is important is supermagnetism or magnetic cluster interaction.

   Your references are quite all and out of the scope of the question related with the spontanous breaking of symmetry and the renormalization group. I think that you needed to take care before to employ so easily terms as "incorrect", because that presumes that you know something of the subject.

  This point is very subtle and you are right that it seems to be out of the context of magnetism which in most of texts is only concerned with the spins, spin-orbit interaction or Hund rules. But if you go to more basic concepts you can see the bands or the states are necessary in the solid, and in fact the Pauli's exclusion principle is what avoids the enormous Weiss local field between magnetic moments and which in fact allows to use the electric energy to act instead of the magnetic one. This is only achieved if you have quantized energies in the solids. Ferromagnetism with only spins could not exist and without states of matter less.

   Finally let me to finish saying that the magnetism in a solid is explained using a Hamiltonian using second quantization operators, i.e. in a Fock space where the quantization of the energy is a granted assumption.