The phase transition from a paramagnetic phase into a ferromagnetic phase is known as an example of spontaneous symmetry breaking. This can be understood by looking at the Heisenberg Hamiltonian which is invariant under rotations. Below T_C a specific alignment of the spin is randomly selected and the symmetry encoded in the Hamiltonian is broken. There is also an analog to the Higgs boson. For a Hamiltonian with continuous rotational symmetry there must be an excitation with vanishing energy in the limit of large wavelength. This mode is known as the Goldstone mode and is the analog to the Higg boson that accompanies the spontaneous symmetry breaking in the high energy scenario. I guess you will find here http://scitation.aip.org/content/aip/magazine/physicstoday/article/66/12/10.1063/PT.3.2212 exactly what you are looking for.

This concept is directly tied to the celebrated Mermin Wagner theorem stating that a Heisenberg ferromagnet can only have a phase transition in 3 spatial dimensions, not in 2 and not in 1. This can be understood intuitively using the spin-wave approach. Here the integral that determines the temperature dependence of the magnetization in the spin-wave approximation diverges due to the Goldstone mode in 1 and 2 dimensions. Only from 3d on there is a meaningful result pointing at the Mermin Wagner theorem.

In high-energy physics, the existence of Higgs boson can explain why some particles have mass. I am wondering what corresponds to the particle mass in condensed matter physics.

Dear Cristian,

all that You write is well known but the sense of my question is different.

It is true, that in condensed matter physics the second order phase transition has a universal concept which is very similar to the Higgs mechanism, based on Landau-Ginsburg type of Hamiltonian. But, note a difference: the interaction is usually mediated by bosons (phonons) which enter in the interaction part of the model Hamiltonian. The resulting gap is determined by these bosons.

Hence, this type of PT is not the object of my question. By the way, the case of the Higgs type: the formation of the gap in where the linear energy dispersion with zero gap at the Dirac point changes to quadratic (that means that mass is generated) and a transition to insulating takes place. This could take place in graphene.

Anderson's idea, should be older, well before graphene.

Anderson's paper is actually Ref. 3 of Higgs' famous PRL, which can be accessed here: http://prl.aps.org/abstract/PRL/v13/i16/p508_1

The reference is: P. W. Anderson, Phys. Rev. 130, 439 (1963)

Dear Malcolm,

thank You very much.

The Anderson reference was in the link I posted, but now you have it anyways.

Christian, thank You very much, I appologize for my overlooking the link. Anyway, end's well, all's well.

When symmetry is broken in condensed matter it has the memory of the symmetry in the Goldstone mode. Is it right? This is the signature in the dynamic property. Can I say for the static property the signature of symmetry breaking is in the domain formation. It is as if the nature tries to compensate the loss of symmetry by producing domains. Can I say that?

Tapan, I do not understand what You mean. What kind of symmetry? In condensed matter as a many body system of interacting entities usually the symmetry breaking is a result of some competing interactions and consequently a change of the ground state. What do You mean by "domains"? Perhaps the domains of ordered spins at magnetic PT ? Goldstone mode is a mode which occurs in very special nonlinear field theoretical models (in condensed matter in low D=1,2 dimensions) ..

Eva,

Below the structural or antiferromagnetic phase transition the symmetry of the lattice changes. The paramagnetic space group has more symmetry elements than the ordered antiferromagnetic phase. It has been observed that the ordered phase has as many antiferromagnetic domains as the number of symmetry elements lost at the phase transition. Thus although the microscopic symmetry is broken there is a tendency that substance retains the macroscopic or global symmetry. This also happens in ferroelectric phase transition. This effect was discussed I guess by Pierre Curie long time ago. Can I call this as the memory effect at the symmetry breaking? I do not know whether this has anything to do with the Anderson's idea of symmetry breaking and imergent phenomena.

Yes, the paramagnetic domains coexist in the ordered ferromagnetic phase as fluctuations below the critical point of the second order PT, but the macroscopic fluctuation is missing there. All the fluctuations vanish only at T=0 where the order parameter grows to its maximum value.

@Tapan,

In my opinion, what you said seems related to decoherence, i.e., the formation of domains obeys the macroscopic physical law, rather than quantum mechanical law. This symmetry-breaking-memory business looks like another language of energy conservation, i.e., the energy saved in the microscopic structure might be dissipated to the environment somehow.

@Wei & Eva: I am an experimentalist and do not have the specialized training to understand the field theoretic papers on condensed matter physics. I know of the great work of Goldstone, Anderson etc. but did not go through them in mathematical details and derivations. But I read the 1972 Science article of P.W. Anderson "More is different" several times and each time I get inspiration from it. The concept of symmetry and its breaking is very fundamental indeed, otherwise we would see a different universe around us.

Yes, indeed. There is a fundamental theorem, the Noether's theorem: Each symmetry implies conservation law of a physical quantity.

Not each symmetry. For example there are geometric symmetries of a

square or a cylinder. There is no conservation related to them.

Noether's theorem needs symmetries of `action' that is

integral of L dt. Where L is the Lagrangian of the system of

point particles under consideration described in terms of

suitable generalized coordinates.

This symmetry operation acts on generalized coordinates. That

is the operation takes one set of generalized coordinates

to another set. If this operation on the generalized coordinates

leaves `action' invariant then that is a symmetry of the `action'.

Next question is that whether this symmetry operation is differentiable.

If it is so then, there is a conserved quantity related to that

symmetry transformation according to Noether's theorem

The comment above is evidently out of the concept of the previous discussion.

The term symmetry was related to a many body system characterized by a Hamiltonian or Lagrangian of certain symmetry, not (surprisingly) to a geometric object, like a square or a cylinder as mentioned above.

The paper by Anderson is called "Plasmons, Gauge invariance and Mass"

Physical Review vol 130 p 439 (1963). There is also a brilliant related paper by Martin Greiter

"Is Electromagnetic Gauge Invariance Spontaneously Violated  in Superconductors?" 

Annals of Physics 319( 2005) p217

This clears up some of the misunderstanding about broken symmetries in superconductors - particularly that which has come from particle physicists.