I had heard that the large radiation belts around Jupiter may be formed by liquid metallic hydrogen in (or around) Jupiter's core (which Wikipedia says haven't been observed in labs yet due to the immense pressure required), but I don't understand how the liquid metallic hydrogen can form a planetary magnetic field. The article also says that the hydrogen can act as a conductor, is this somehow related to moving electric fields forming magnetic fields?
As you stated, we have not been able to simulate the pressure and temperatures required to generate those that are believed to exist in Jupiter's interior other than in short-lived shockwave experiments, according to the NASA webpage A Freaky Fluid inside Jupiter?, observing that
"Liquid metallic hydrogen has low viscosity, like water, and it's a good electrical and thermal conductor," says Caltech's David Stevenson, an expert in planet formation, evolution, and structure. "Like a mirror, it reflects light, so if you were immersed in it [here's hoping you never are], you wouldn't be able to see anything."
Going further, according to the article Jumpin' Jupiter! Metallic Hydrogen (Lawrence Livermore National Laboratory), discuss the shock wave results, finding the level at which hydrogen metallises as being
from 0.9 to 1.4 Mbar, resistivity in the shocked fluid decreases almost four orders of magnitude (i.e., conductivity increases); from 1.4 to 1.8 Mbar, resistivity is essentially constant at a value typical of that of liquid metals. Our data indicate a continuous transition from a semiconducting to metallic diatomic fluid at 1.4 Mbar, nine-fold compression of initial liquid density, and 3,000 K.
The findings from the researchers above are summarised in the diagram below
The source is the Jumping Jupiter link above.
This is just an amusing addition to the existing answer.
It turns out that a metallic hydrogen layer (which lets electrons move freely, and moving electrons means a magnetic field can form) is not enough to account for the size of Jupiter's magnetosphere. It's off by a factor of approximately 2.
The rest of it is mostly thanks to Io. The wiki page will give a more complete description (of a rather complex system) and references, but here's the short of the matter.
Io is on an eccentric orbit, thanks to a resonance with the other Galilean moons. This gives it significant tidal heating (and as it is the closest of the Galilean moons, it has the most significant heating effect of them all). This gives it volcanic activity, which puts new gaseous materials (sulfur, oxygen, and chlorine mostly) into its atmosphere. Jupiter strips material from Io's upper atmosphere at about 1 metric ton per second. This material ultimately forms ionized bands that generate substantial electric current, and significantly increases the magnetosphere around Jupiter as a result.
The mechanism that produces Jupiter's magnetic field is in principle not any different to that which produces the earth's. Only that in the latter case it works with liquid iron instead of hydrogen. You should read about the dynamo theory https://en.wikipedia.org/wiki/Dynamo_theory