Spectral gap (physics)

{{Short description|Energy difference between ground and first excited states}}

In quantum mechanics, the spectral gap of a system is the energy difference between its ground state and its first excited state.{{Cite journal |title=Undecidability of the spectral gap |journal = Nature|volume = 528|issue = 7581|last1=Cubitt |first1=Toby S. |last2=Perez-Garcia |first2=David |date=2015-12-10 |location=US |pages=207–211 |language=en-us |doi=10.1038/nature16059 |pmid=26659181 |last3=Wolf |first3=Michael M.|arxiv = 1502.04135 |bibcode=2015Natur.528..207C|s2cid = 4451987}}{{cite web |url= https://futurism.com/19474|title=Scientists Just Proved A Fundamental Quantum Physics Problem is Unsolvable|last=Lim|first=Jappy|date=11 December 2015|website=Futurism|access-date=18 December 2018}} 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.{{cite journal |last1=Nachtergaele |first1=Bruno |last2=Sims |first2=Robert |title=Lieb-Robinson Bounds and the Exponential Clustering Theorem |journal=Communications in Mathematical Physics |date=22 March 2006 |volume=265 |issue=1 |pages=119–130 |doi=10.1007/s00220-006-1556-1|arxiv=math-ph/0506030 |bibcode=2006CMaPh.265..119N |s2cid=815023 }}{{cite journal |last1=Hastings |first1=Matthew B. |last2=Koma |first2=Tohru |title=Spectral Gap and Exponential Decay of Correlations |journal=Communications in Mathematical Physics |date=22 April 2006 |volume=265 |issue=3 |pages=781–804 |doi=10.1007/s00220-006-0030-4|arxiv=math-ph/0507008 |bibcode=2006CMaPh.265..781H |s2cid=7941730 }}{{cite journal |last1=Gosset |first1=David |last2=Huang |first2=Yichen |title=Correlation Length versus Gap in Frustration-Free Systems |journal=Physical Review Letters |date=3 March 2016 |volume=116 |issue=9 |page=097202 |doi=10.1103/PhysRevLett.116.097202|pmid=26991196 |arxiv=1509.06360 |bibcode=2016PhRvL.116i7202G |doi-access=free }}

In 2015, it was shown that the problem of determining the existence of a spectral gap is undecidable in two or more dimensions.{{Cite journal | doi=10.1038/nature16059|pmid = 26659181| title=Undecidability of the spectral gap| journal=Nature| volume=528| issue=7581| pages=207–211| year=2015| last1=Cubitt| first1=Toby S.| last2=Perez-Garcia| first2=David| last3=Wolf| first3=Michael M.|bibcode = 2015Natur.528..207C|arxiv = 1502.04135|s2cid = 4451987}}{{cite journal |last1=Kreinovich|first1=Vladik|title=Why Some Physicists Are Excited About the Undecidability of the Spectral Gap Problem and Why Should We|url=https://digitalcommons.utep.edu/cgi/viewcontent.cgi?article=2163&context=cs_techrep|journal=Bulletin of the European Association for Theoretical Computer Science|volume=122|issue=2017|access-date=18 December 2018}} 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.{{cite magazine |last1=Cubitt|first1=Toby S.|last2=Perez-Garcia|first2=David|last3=Wolf|first3=Michael M.|date=November 2018|title=The Unsolvable Problem|url=https://www.scientificamerican.com/article/the-unsolvable-problem/|magazine=Scientific American|url-access=subscription}} 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.{{cite journal |last1=Bausch |first1=Johannes |last2=Cubitt |first2=Toby S. |last3=Lucia |first3=Angelo |last4=Perez-Garcia |first4=David |title=Undecidability of the Spectral Gap in One Dimension |journal=Physical Review X |date=17 August 2020 |volume=10 |issue=3 |pages=031038 |doi=10.1103/PhysRevX.10.031038 |bibcode=2020PhRvX..10c1038B |s2cid=73583883 |doi-access=free |arxiv=1810.01858 }}