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It seems that the phase velocity of light could be greater than $c$, if $\sqrt{\epsilon \mu} < 1/c$, i.e. for anomalous dispersion.

  1. Are there examples of such media? For diamagnetics it seems possible since $\mu < \mu_0$.

  2. And what is the physical meaning of phase velocity in this case, besides being speed of wave surface? Why it cannot be seen as a physical entity that is traveling faster than light?

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Related question –  Vijay Murthy Apr 6 '12 at 14:00
    
This is the historical motivation for the Kramers-Kronig dispersion relation, which completely explained causality with a varying complex dielectric constant and susceptibility, and replaced causality in S-matrix theory. –  Ron Maimon Apr 7 '12 at 4:25
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Re your question 2: this is how I like to think about it.

If you have an infinitely long steady light beam it cannot transmit any information, so it doesn't really have a speed. The wave has a phase velocity, but this is just the speed that the phase of the wave moves and this doen't trasmit any information. To send information with a light beam you need to modulate it. You could just pulse it on and off, like Morse code, or you could modulate it's amplitude like AM radio. The speed these changes in the beam amplitude travel is called the group velocity, and while the group velocity can be equal to the phase velocity it doesn't have to be. It's the group velocity that moves at or below the speed of light. The phase velocity can move arbitrarily fast without violating special relativity.

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