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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 $k$.

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The term "massless" originates in relativistic quantum field theory, where mass-energy equivalence means that an energy gap in the Hamiltonian's spectrum is equivalent to a mass for the corresponding particle.

Condensed-matter systems can indeed have a gapless quadratic dispersion relation, as you point out. But such a system is inherently non-relativistic; it does not possess Lorentz invariance. (In particular, it is scale invariant but not conformally symmetric, and it cannot be described by a CFT.) So it doesn't really make sense to talk about the "mass" of the excitations in such a system; but only about their energy gap.

So strictly speaking, the term "massless" is simply meaningless in reference to such a system. However, some people sloppily refer to it as "massless" by analogy with relativistic systems, in which case they really mean "gapless".

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