Sunday 16 August 2015

Is every spontaneous symmetry breaking connected with a second order phase transition?



My question is rather clear: Is every spontaneous symmetry breaking connected with a second order phase transition?


In many books (A.Zee QFT in a nutshell or also in P&S etc.) when they start explaining spontaneous symmetry breaking, the case of loss of ferromagnetism at the Curie temperature is mentioned as an example. This transition from a state with non-zero directed magnetisation to zero magnetisation is a well-known example of second-order phase transition. Bose-Einstein condensation is also considered as a second order phase transition. A.Zee describes in his book QFT in a nutshell that superfluidity is a process of spontaneous symmetry breaking with a non-relativistic boson-field with Mexican-hat potential and explains even the emergence of Nambu-Goldstone bosons in case of the symmetry breaking. I assume that in case of a gauge field symmetry breaking the situation might be a bit different.


It is also rather remarkable that according to the Landau-theory of second-order transitions the dependence of Gibbs potential on the magnetisation looks very much like a Mexican potential in one dimension (see fig. 8.3 of P&S) for the case $T


Answer



The answer above is great example. I would give another simple situation.


If the system have $Z_2$ symmetry, and the free energy could be written in the form: $$f(m)= a(T-T_c)m^2+ b m^4$$ then as we cool the system from higher temperature to below $T_c$, there is a spontaneous symmetry breaking and second order phase transition;


However, if the $b<0$, then we have to consider $m^6$ term: $$f(m)= a(T-T_c)m^2 - |b| m^4 +c m^6$$


in this case, as we cool the system from higher temperature, before we reach $T_c$, two other local minimums of free energy appear(Fig(1)); as we further cool the system, these two local minimums become the global minimums(Fig(2)); but this case is a first order phase transition. Fig(1)


Fig(2)


No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...