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] mass gap is the spectral gap between the vacuum and the lightest particle. A Hamiltonian with a spectral gap is called a gapped Hamiltoniangapless .
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] aperiodic tiling of quantum Turing machines and showed that this hypothetical material becomes gapped if and only if the machine halts.[ 8] 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 .
