I would like to ask what it means exactly for a many-body system to have a nonzero $n$-particle gap. If I have a spectrum for each number sector of the Hilbert space for a number conserving Hamiltonian, how is a single particle, pair or $n$-particle excitation defined generically? And what gap are we targeting (which energy difference)? I am trying to make sense of statements such as "The system has gapped single particle excitations and gapless pair excitations".
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The gap refers to the energy gap between the lowest energy eigenvalue in the $n$-particle sector and the absolute lowest energy eigenvalue across all sectors (which, for a stable Hamiltonian that’s bounded below, lies in the zero-particle vacuum sector of the Hilbert space).
So the statement “The system has gapped single particle excitations and gapless pair excitations" means that the lowest energy eigenvalues in the zero- and two-particle sectors are degenerate (in the infinite-system-size limit), while the lowest energy eigenvalue in the one-particle sector is strictly higher.
