What is meant by spontaneous symmetry breaking? What is meant by spontaneous symmetry breaking? I am really confused about this topic. In my text book it says:

"At a continuous phase transition, the slope of the free energy curve changes continuously and a symmetry is always broken".

What symmetry is broken? Why is it broken? How can I see that it is broken? Can someone explain the logic in this pls?
 A: Since it seems that you're dealing with this in a statistical mechanics context, I'll use the Ising model as an example.
Spontaneous symmetry breaking is a phenomenon in which the Hamiltonian or Lagrangian of your system has a certain symmetry, but some relevant state of your system does not. Take the nearest neighbor 2D Ising model
$$ H=-J\sum_{\langle i,j\rangle} s_is_j$$
It's clear that this Hamiltonian has a spin inversion symmetry ($\mathbb Z_2$ symmetry), i.e. if you flip all the spins, $s_i\to -s_i$, the Hamiltonian doesn't change, so that for example an all up configuration has the same energy as an all down configuration. This would lead you to think that the magnetization
$$ \mu=\frac{1}{N}\sum_i s_i$$
has $0$ thermal average at every temperature, because the configuration that gives rise to a magnetization $\mu$ has the same exact probability of showing up as the configuration that has magnetization $-\mu$, and the two cancel. Nevertheless, in the thermodynamic limit, spontaneous symmetry breaking happens. (Source of the picture)

as you can see, at high temperature the mean magnetization is $0$, as expected, but at low temperature the symmetry is broken and the system is forced into a state that has either mostly up spins or mostly down spins with finite magnetization, hence while the Hamiltonian has spin flip symmetry, the state doesn't.
Intuitively, the $0$ temperature state is one with either all spin up or all spin down, so it is easy to see that at low temperature there are two disconnected entropy maximizing states, disconnected in the sense that there is a large energy barrier between them. As you lower the temperature the system has to "pick" one of the two and break the symmetry.
This is where the concept of an "order parameter" for a phase transition comes from. An order parameter is a quantity, in this case $\mu$, that vanishes on average if the state has all the symmetries of the Hamiltonian. As you cross a critical temperature, you will see order parameters taking finite values, which signals spontaneous symmetry breaking.
A: I will try a simple explanation. Try putting a cylindrical pencil standing upright on a ball. Eventually, for a second, you will succeed, but then the pencil will probably fall.
What tells the pencil in which direction to fall? At the start (pencil upright on the ball) there is cylindrical symmetry to the whole thing. The answer would be that some fluctuations, say butterfly flapping, will eventually cause some tilt in some direction, and then the negative feedback due to gravity will drive it the rest of the way. Thus a small effect causes a very large one.
More commonly I heard about symmetry breaking in magnets, that spontaneously magnetize below Curie Temperature. Again, in which direction should it magnetize? Small noise somewhere, and then the chain reaction of magnetic domains rearranging does the rest. Another example water freezing. Where should it start freezing first? Again....
