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.<ref>Cubitt, Toby S.; Perez-Garcia, David; Wolf, Michael M. (2015-12-10). "Undecidability of the spectral gap". Nature. US. 528 (7581): 207–211. arXiv:1502.04135. Bibcode:2015Natur.528..207C. doi:10.1038/nature16059. PMID 26659181. S2CID 4451987.</ref><ref>Lim, Jappy (11 December 2015). "Scientists Just Proved A Fundamental Quantum Physics Problem is Unsolvable". Futurism. Retrieved 18 December 2018.</ref> 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.<ref>Nachtergaele, Bruno; Sims, Robert (22 March 2006). "Lieb-Robinson Bounds and the Exponential Clustering Theorem". Communications in Mathematical Physics. 265 (1): 119–130. arXiv:math-ph/0506030. Bibcode:2006CMaPh.265..119N. doi:10.1007/s00220-006-1556-1. S2CID 815023.</ref><ref>Hastings, Matthew B.; Koma, Tohru (22 April 2006). "Spectral Gap and Exponential Decay of Correlations". Communications in Mathematical Physics. 265 (3): 781–804. arXiv:math-ph/0507008. Bibcode:2006CMaPh.265..781H. doi:10.1007/s00220-006-0030-4. S2CID 7941730.</ref><ref>Gosset, David; Huang, Yichen (3 March 2016). "Correlation Length versus Gap in Frustration-Free Systems". Physical Review Letters. 116 (9): 097202. arXiv:1509.06360. Bibcode:2016PhRvL.116i7202G. doi:10.1103/PhysRevLett.116.097202. PMID 26991196.</ref>

In 2015, it was shown that the problem of determining the existence of a spectral gap is undecidable in two or more dimensions.<ref name="proof">Cubitt, Toby S.; Perez-Garcia, David; Wolf, Michael M. (2015). "Undecidability of the spectral gap". Nature. 528 (7581): 207–211. arXiv:1502.04135. Bibcode:2015Natur.528..207C. doi:10.1038/nature16059. PMID 26659181. S2CID 4451987.</ref><ref name="kreinovich">Kreinovich, Vladik. "Why Some Physicists Are Excited About the Undecidability of the Spectral Gap Problem and Why Should We". Bulletin of the European Association for Theoretical Computer Science. 122 (2017). Retrieved 18 December 2018.</ref> 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.<ref>Cubitt, Toby S.; Perez-Garcia, David; Wolf, Michael M. (November 2018). "The Unsolvable Problem". Scientific American.</ref> 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.<ref>Bausch, Johannes; Cubitt, Toby S.; Lucia, Angelo; Perez-Garcia, David (17 August 2020). "Undecidability of the Spectral Gap in One Dimension". Physical Review X. 10 (3): 031038. arXiv:1810.01858. Bibcode:2020PhRvX..10c1038B. doi:10.1103/PhysRevX.10.031038. S2CID 73583883.</ref>

See also

• List of undecidable problems
• Spectral gap, in mathematics