Can a medium can have refractive index less than the reference refractive index (that is less than 1)? I know taking the refractive index of the vacuum as unity, the refractive index of all the objects is calculated. So my question is that is it possible that a medium can have refractive index less than the reference refractive index (that is less than 1)? If yes, then what is its physical significance?
 A: As in VictorSeven's answer, the refractive index can be less than one, but its frequency dependence in so doing must be such that effects cannot travel through the medium in question at faster than the universal signalling speed limit $c$. An electromagnetic input to the quiescent medium cannot lead to nonzero electromagnetic disturbance at a point $d$ distant from the point of input before a time $d/c$ has elapsed after the beginning of the input.
As also in VictorSeven's answer, an approximate restatement of this rather wordy assertion is that the medium respects the universal signalling speed limit iff the group velocity is less than $c$. But even this is an approximation, albeit almost always an excellent one. The only non-approximate criterion is to calculate the linear electromagnetic impulse response of a thickness $d$ of the medium in question and to check that this latter response is nought for all times before $d/c$ seconds has elapsed after the input impulse. So it's not possible to give simple, failsafe criteria for the refractive index, one can only formulate criteria about the Fourier transform of the complex refractive index. Amongst other things, this means that the magnitude of the complex refractive index must fulfill the Payley-Wiener criterion and Payley-Wiener theorem.
A: Absolute refractive index of a medium is speed of light in vacuum / speed of light in in that medium.
This will always be greater than one as speed of light decreases as density increases and vacuum is rarer(opposite of dense) than anything that we know of.
However, there are some cases where refractive index is less than 1 and also negative for some. These are like those few exceptions which are different from majority and occur in some rare cases only. Like RI of water for x-rays is below 1.
If a material like this exists which has a RI less than one despite some Exception like condition, then it would wrong many theories of physics and change a lot of our understanding about the universe.
A: Yes, you can have refractive indices less than one.
This is because the theory of relativity limits the group velocity to be less than $c$. Group velocity is the spèed at which information travels, and can be computed as:
$$v_g = \frac{d\omega}{dk}$$
where $\omega$ is the frequency and $k$ the wave number. In a non-dispersive medium, $\omega = k v_p$ and then $v_g = v_p$.
$v_p$ is the phase velocity, at which the perturbation of the wave travels. The index of refraction is defined using this speed: $$n=\frac{c}{v_p}$$
Since most times the relation $\omega = k v_p$ holds, both velocities (phase and group) are the same, so we cannot have $v_p$ greater than $c$ and the index of refraction is usually larger than 1.
However, the thing is that in some cases, we can have $v_p > c$ (always with $v_g < c$). This leads to rare cases where we can have a refractive index which less than 1, without violating special relativity.
It is also in the Wiki: https://en.wikipedia.org/wiki/Refractive_index#Refractive_index_below_unity
