# What's the definition of "massive" and "massless" in the condensed matter physics (CMP)?

Initially, I think "massless" equals to "gapless spectrum", and "massive" equals to "gapped spectrum". This is because of the example of Altland's book P199-200, which says:

if Green function can be written as:$$G=\frac{1}{p^2+m}=\frac{1}{-\omega^2 +k^2+m}$$ the dispersion is $$\omega(k)=\sqrt{k^2+m}$$

As the result, we can say, if $$m\rightarrow 0$$, i.e. massless, the spectrum is gapless and is linear at small $$k$$, also, we can even say the quasi-particle can long-range propagate since the correlation length $$\xi=m^{-1/2}$$. On the hands, if $$m\neq 0$$, i.e. massive, the spectrum is gapped and is quadratic at small $$k$$, also, the quasi-particle can only short-range propagate.

What I am confused is that there are some quadratic dispersion but gapless, e.g. magnon of FM, but some books argue it is massive excitation, which is different from massless excitation of AFM. This kind of dispersion is corresponding to another form of Green function: $$G\sim\frac{1}{-\omega +k^2}$$.

So I want to know what the definition of "massive excitation" in the CMP?

By the way, I am also confused the hidden reason of such distinction between magnon dispersion of AFM and FM, i.e. linear v.s. quadratic at small $$k$$.