Topological Phase Transition v Quantum Phase Transition v Phase Transition

Question

What are the main differences between this 3 type of phase transition?

I understand the phase transitions of gas/liquid/solid as well as ferromagnet/paramagnet(Ising Model). All of which are between ordered and disordered state.

But do all of the transitions are between ordered and disordered state? Between low entropy state and high entropy state? If it is, what is the topological order here? The number of chiral-spin/vortex/antivortex?

Based on some googling, it seems like both occurs at a very low temperature (0K if not mistaken) which neglect the temperature fluctuation. So in this case what is the parameters/variables that are tuned to see the transition?

Answer 1

Phase transition in system behaviour, governed by some control parameters T, means drastic change in behaviour of the system when T vary. But how"drastic change" may occur if usually dynamics of the physical systems are described by differential equations?

First we have to rule out situations where differential equations of dynamics of the system has discontinuity in their coefficients. In such circumstances, discontinuity of the solutions and drastic change is just caused by coefficients of the equations.

Second we should rule out various phenomena like shock waves etc. They are caused by some other mechanisms, however their description may be similar in some ways to those of phase change.

What remains, usually has the following reason. At typical, physical systems has defined phase spaces. It is a space of momentum and velocity, and typical system dynamics may went through any point of this space, given enough energy and properly crafted initial conditions. For many particle systems, evolution in phase space is completely homogeneous. But in some circumstances, mainly when dynamics is bounded by symmetry of the system, certain parts of the phase space may be not accessible for the system below some value of the control parameter T. And is accessible above. In this case when T cross certain value, dynamics drastically change, as there is change in accessible phase space volume, which is usually drastic ( for example small part vs. all phase space).

In this description it may be symmetry braking due to lowering temperature, or it may be topological invariant which is conserved as long as certain terms in equations are dominant.

Various phase changes has very different physical mechanisms underlying, but due to universality hypothesis, they are in fact similar from mathematical point of view, and grouped in various kinds depending on discontinuity which appears in a the a mechanism of a change and tensor rank of the order parameter. In all cases change of the accessible phase space, due to various mechanisms, is the reason.