Spectral gap (physics)

In quantum mechanics, the spectral gap of a system is the energy difference between its ground state and its first excited state.^[1][2] The mass gap is the spectral gap between the vacuum and the lightest particle. A Hamiltonian with a spectral gap is called a gapped Hamiltonian, and those that do not are called gapless.

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.^[9]

See also

• List of undecidable problems
• Spectral gap, in mathematics