quantum mechanics - What does it mean for a Hamiltonian or system to be gapped or gapless?

I've read some papers recently that talk about gapped Hamiltonians or gapless systems, but what does it mean?

Edit: Is an XX spin chain in a magnetic field gapped? Why or why not?

Answer

Gapped or gapless is a distinction between continuous and discrete spectra of low energy excitations. For a Hamiltonian $H$ with gapped spectrum, the first excited state has an energy eigenvalue $E_1$ that is separated by a gap $\Delta > 0$ from the ground state $E_0$. For example, a dispersion relation of the form $E = |k|$ is an example of a gapless (continuous) spectrum, whereas $E = \sqrt{k^2 + m^2}$ is an example of a gapped one. $k$ denotes the wave vector and can be any real number. $m$ is the mass which in this case is the cause of the gap.

This distinction leads to a qualitative difference in the physical behavior of gapped and untapped systems - most importantly it determines whether a material is a conductor or an insulator. There are quite fascinating processes that can give rise to a gap such as interactions (interesting examples are the mass gap in Yang-Mills theory, or the gap in BCS superconductivity).