What are phase transiton in different contexts?

A very large class of phase transitions are characterized by the breaking of some symmetry. Usually one finds a quantity called the order parameter, and finds its scaling with respect to an energy scale for the system, like the temperature. Usually for a phase transition one finds that the order parameter is either discontinuous or one of its derivatives is discontinuous. Now finding an order parameter is more of an art than a procedure, so a more systematic way of studying phase transitions is within the frame work of Landau theory.

In this framework one studies the free energy of the system in terms of an averaged or course grained version of the system. The main assumption is that the free energy can be expanded in terms of a ‘mean field" $phi$

begin{equation} mathcal{F}(phi)= a_1 (partial phi)^2+sum_n b_n phi^n + dots end{equation} Where the coefficients are found empirically at a particular temperature. As one course grains the system more and more we get different effective descriptions at each step because the coefficients $a_i$, $b_i$ change (this is in effect changing the energy scale at which the system sits). This means that at different scales the graph of the free energy looks different and roughly has different symmetries. The most famous example of this is the "mexican hat" potential.

begin{equation} V(phi)= mu phi^2 + lambda phi^4 end{equation} When $mu>0$ this potential has one minimum at $phi=0$, but as $mu$ becomes negative the potential has two minima. In the case where there is one minimum to the potential, small fluctuations around this potential are stable and have a symmetry under change of sign $delta phi rightarrow -delta phi$. When there are two minima this is no longer the case becase changing the sign takes you from one vacuum to the other, so in that sense the symmetry is "spontaneously broken".

This whole procedure is usually called the Renormalization group. Critical points of a system are described by fixed points of the RG flow, meaning that they are scale invariant and the correlation length of excitations becomes infinite. This whole formalism is basically the same as a (Wilsonian) quantum field theory. So actually studying a big class of phase transitions (the ones that come from breaking of a symmetry) is equivalent to studying quantum field theories, and the study of critical points is the study of what are called conformal field theories.

Now it actually turns out that there are some phase transitions that are not characterized by the breaking of any symmetry. A good example of a real world example of this are topological insulators. In this case the phase is characterized by some "topological charge". There is a mathematical framework to try to classify all such phases of matter in terms of some objects called (braided) tensor categories.

For me a phase transition is defined (somewhat strictly) as a non-analyticity in the thermodynamic potential. A function $f(x)$ is non-analytic at $x=x_0$ if it can not be expanded in powers of $x-x_0$. For example $f(x) = sqrt{x+1}$ can be expanded around $x=0$

$$f(x) cong 1 + frac{x}{2} - frac{x^2}{8} + dots $$

but this does not work at $x=-1$ because the first derivative of $f(x)$ diverges at $x=-1$. A more realistic example would be a function like $f(x) = sqrt{x^2+m^2}$ with $m$ to be interpreted as some parameter of the system. We see that $f(x)$ is analytic for all values of $x$, except when $m=0$ and then we get a cusp $f(x) = left|xright|$. This means that there is a special point in the phase diagram where the free energy is non-analytic. This is a phase transition.

For example, in the Ising model, the free energy is a continuous function of the magnetic field $h$, but is has a cusp at $h=0$. Then the magnetisation (which is the derivative of the free energy with respect to the magnetic field) is discontinuous. At a the critical point, the cusp goes away but the magnetisation behaves as a power law (with a non-integer exponent). A different non-analyticity emerges.

I like this definition because it is directly tied to the fact that something special happens at phase transitions. Indeed, in statistical physics the Free energy is computed by taking the logarithm of the partition function

$$ F = -k_{text{B}} T , text{Ln}left(sum_{text{states}} text{e}^{-E_{text{state}}/(k_{text{B}T})}right) , .$$

This is a sum of finite elements. Then it is analytic almost by construction. The way out is that the system may contain infinitely many particles so that the sum has infinitely many elements. Think for example of the geometric series, $sum_{n=0}^infty x^n = 1/(1-x)$ which is not analytic at $x=1$ even though this does not appear in the individual elements of the sum. The finite sum $sum_{n=0}^N x^n = (1-x^{N+1})/(1-x)$, is always perfectly analytic. Physically, this means that a phase transition can only occur in the thermodynamic limit. Systems without an infinity somwhere do not have phase transitions.

This is of course a bit strict and needs to be generalised some times. The idea however remains. For example

• Dynamical phase transitions can emerge when we look at the state of a system after an infinite time evolution as a function of its microscopic parameters.
• In Renormalisation Group (RG) methods, we purposefully avoid non-analiticities by introducing a cut-off, but we let the cut-off run to infinity.
• In quantum phase transitions, we send the temperature to zero (see the exponentials in the above equation where $1/T$ goes to infinity).
• In the Ginzburg-Laudau picture, the free energy is first given as an analytic function of the magnetic field and the magnetisation. The physical free energy depends however only on the magnetic field with the magnetisation being computed by looking for the minimum of the free energy. This minimisation produces the non-analyticity and is equivalent to finding the asymptotic equilibrium of a dynamical system.