nLab spectral gap

Idea

Generally

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

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).

While spectral gaps are ubiquituous in single-atom quantum systems (cf. the famous discrete energy levels of the hydrogen atom) they tend to disappear in the thermodynamic limit when many atoms are brought close together, whereby their energy levels fuse into continuous energy bands.

For this reason, in condensed matter theory “gapped Hamiltonians” are the remarkable exception, not the rule (cf. also topological phases of matter).

Related concepts

• gapped Hamiltonian

• quantum adiabatic theorem

• topological phase of matter

• Yang-Mills 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)

On rigorous results about lattice (spin) models which remain gapped in the continuum limit:

• Houssam Abdul-Rahman, Marius Lemm, Angelo Lucia, Bruno Nachtergaele, Amanda Young, A class of two-dimensional AKLT models with a gap, in: Analytic Trends in Mathematical Physics, Contemporary Mathematics 741, AMS (2020) [arXiv:1901.09297, doi:10.1090/conm/741/14917]

• Simone Warzel, Amanda Young: A Bulk Spectral Gap in the Presence of Edge States for a Truncated Pseudopotential, Ann. Henri Poincaré 24 (2023) 133–178 [arXiv:2108.10794, doi:10.1007/s00023-022-01210-z]

• Angelo Lucia, Amanda Young: A Nonvanishing Spectral Gap for AKLT Models on Generalized Decorated Graphs, J. Math. Phys. 64 (2023) 041902 [arXiv:2212.11872, doi:10.1063/5.0139706]

• Amanda Young: On a bulk gap strategy for quantum lattice models, Reviews in Mathematical Physics 36 09 (2024) 2460007 [arXiv:2308.01405, doi:10.1142/S0129055X24600079]

• Amanda Young: Quantum Spin Systems [arXiv:2308.07848, spire:2688309]

Review:

• Amanda Young: Gap stability and the AKLT model on the hexagonal lattice, talk at QMATH16, Munich (2025) [pdf]