Physics:Spectral gap
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
References
- ↑ Cubitt, Toby S.; Perez-Garcia, David; Wolf, Michael M. (2015-12-10). "Undecidability of the spectral gap" (in en-us). Nature (US) 528 (7581): 207–211. doi:10.1038/nature16059. PMID 26659181. Bibcode: 2015Natur.528..207C.
- ↑ Lim, Jappy (11 December 2015). "Scientists Just Proved A Fundamental Quantum Physics Problem is Unsolvable". https://futurism.com/19474.
- ↑ Nachtergaele, Bruno; Sims, Robert (22 March 2006). "Lieb-Robinson Bounds and the Exponential Clustering Theorem". Communications in Mathematical Physics 265 (1): 119–130. doi:10.1007/s00220-006-1556-1. Bibcode: 2006CMaPh.265..119N.
- ↑ Hastings, Matthew B.; Koma, Tohru (22 April 2006). "Spectral Gap and Exponential Decay of Correlations". Communications in Mathematical Physics 265 (3): 781–804. doi:10.1007/s00220-006-0030-4. Bibcode: 2006CMaPh.265..781H.
- ↑ Gosset, David; Huang, Yichen (3 March 2016). "Correlation Length versus Gap in Frustration-Free Systems". Physical Review Letters 116 (9): 097202. doi:10.1103/PhysRevLett.116.097202. PMID 26991196. Bibcode: 2016PhRvL.116i7202G.
- ↑ Cubitt, Toby S.; Perez-Garcia, David; Wolf, Michael M. (2015). "Undecidability of the spectral gap". Nature 528 (7581): 207–211. doi:10.1038/nature16059. PMID 26659181. Bibcode: 2015Natur.528..207C.
- ↑ 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). https://digitalcommons.utep.edu/cgi/viewcontent.cgi?article=2163&context=cs_techrep. Retrieved 18 December 2018.
- ↑ Cubitt, Toby S.; Perez-Garcia, David; Wolf, Michael M. (November 2018). "The Unsolvable Problem". Scientific American. https://www.scientificamerican.com/article/the-unsolvable-problem/.
- ↑ 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. doi:10.1103/PhysRevX.10.031038. Bibcode: 2020PhRvX..10c1038B.
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