Red Giant Stars

As the inert He core contracts, regions of the star just outside the core heat up as well, until they reach temperatures where they can fuse H -> He. Hydrogen shell burning commences:

Inside the core:

This shell actually produces more energy than the normal hydrogen burning core phase did, so luminosity rises.
Not all the energy escapes; some goes into expanding the star: radius rises, surface temperature drops.
The star becomes a subgiant star.

As more and more helium "ash" is being added to the core, it keeps contracting to maintain pressure and hold up the star. Once the amount of mass in the helium core reaches about 8% of the star mass (the Schonberg-Chandrasekhar Limit), the density becomes so high that the core no longer acts like a perfect gas. It becomes degenerate.

Electron degeneracy - back to quantum mechanics!

Electrons obey a quantum mechanical principle known as the Pauli Exclusion Principle: no two electrons can share the same quantum state.
What does this mean? Electrons want to sit in the lowest possible energy state. They can't all do that. So the higher the density, the more the electrons have to exist in higher and higher energy states. More energy --> more motion --> higher pressure.
If the gas is completely degenerate, the follow an equation of state which is given by
Note: no temperature dependence!

So we have:

an inert helium core supported by electron degeneracy,
a hydrogen burning shell
an expanding outer envelope

This leads to...

The Red Giant Ascent

As the star expands, it cools. At this point, the H^- ion can form, when a neutral hydrogen atom "grabs" a free electrons (from partial ionization of other elements). It is very easy to for radiation to be absorbed by, and ionize, the H^- ion -- so opacity is high.

High opacity + high energy generation = convection! The whole outer envelope becomes convective, and processed material in the core can rise to the surface, a process known as dredge-up.

As the outer layers continue to expand, ionization drops so there are fewer free electrons and fewer H^- ions. Opacity drops and luminosity keeps increasing (due to core contraction).

The star becomes a red giant:

L > 100 L_sun
T_e ~ 4000 K
R ~ 30 - 100 R_sun
examples: Arcturus & Aldebaran

Meanwhile, down in the core...

...temperature keeps rising. When the temperature gets high enough (100 million degrees), Helium can begin to fuse together to form Carbon in the triple alpha process.

but the core is degenerate: so that when temperature rises, pressure does not: the core can't expand and cool... so energy raises temperature raises energy raises temperature raises energy raises temperature: a runaway!

When the triple alpha process kicks in, 10¹¹ L_sun is released in a few seconds: the helium core flash.

None of the energy makes it out of the star. Instead it lifts the electron degeneracy (making it behave again like an ideal gas) and expands the core a bit. So we now have:

a helium burning core
a hydrogen burning shell (providing most of the luminosity)
an extended envelope

but since the core expands, it cools and the energy generation in the hydrogen burning shell drops. The luminosity decreases, the star shrinks, and the surface temperature rises. The star becomes a horizontal branch star.