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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 .
[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
References
^ Cubitt, Toby S.; Perez-Garcia, David; Wolf, Michael M. (2015-12-10). "Undecidability of the spectral gap". Nature . 528 (7581). US: 207–211.
arXiv :
1502.04135 .
Bibcode :
2015Natur.528..207C .
doi :
10.1038/nature16059 .
PMID
26659181 .
S2CID
4451987 .
^ Lim, Jappy (11 December 2015).
"Scientists Just Proved A Fundamental Quantum Physics Problem is Unsolvable" . Futurism . Retrieved 18 December 2018 .
^ 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 .
^ 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 .
^ 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 .
^ 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 .
^ 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 .
^ Cubitt, Toby S.; Perez-Garcia, David; Wolf, Michael M. (November 2018).
"The Unsolvable Problem" . Scientific American .
^ 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 .