nLab spectral gap

Idea

Generally, given a (complex) linear operator on a separable Hilbert space with real operator spectrum, then a spectral gap is any interval $I \subset \mathbb{R}$ not intersecting the spectrum.

In quantum physics this is often considered for the Hamiltonian operator of a quantum system, whence one also speaks of an “energy gap”.

Moreover, often the term is by default understood as referring to a gap above the ground state, hence such that the only energy eigenvalue smaller than all $E \in I$ is that of the ground state (typically taken to be zero).

Related concepts

• gapped Hamiltonian

• quantum adiabatic theorem

• topological phase of matter

• mass gap

References

• Toby Cubitt, David Perez-Garcia, Michael M. Wolf: Undecidability of the Spectral Gap, Forum of Mathematics, Pi 10 (2022) e14 [doi:10.1017/fmp.2021.15]

See also:

• Wikipedia: Spectral gap (physics)