In solid-state physics, the most important spectral gap is for the many-body system of electrons in a solid material, in which case it is often known as an energy gap.
In quantum many-body systems, ground states of gapped Hamiltonians have exponential decay of correlations.
[3][4][5] In 2015, it was shown that the problem of determining the existence of a spectral gap is undecidable in two or more dimensions.
[6][7] The authors used an aperiodic tiling of quantum Turing machines and showed that this hypothetical material becomes gapped if and only if the machine halts.
[8] The one-dimensional case was also proven undecidable in 2020 by constructing a chain of interacting qudits divided into blocks that gain energy if and only if they represent a full computation by a Turing machine, and showing that this system becomes gapped if and only if the machine does not halt.