Gapped Hamiltonian

In many-body physics, most commonly within condensed-matter physics, a gapped Hamiltonian is a Hamiltonian for an infinitely large many-body system where there is a finite energy gap separating the (possibly degenerate) ground space from the first excited states. A Hamiltonian that is not gapped is called gapless.

The property of being gapped or gapless is formally defined through a sequence of Hamiltonians on finite lattices in the thermodynamic limit.[1]

An example is the BCS Hamiltonian in the theory of superconductivity.

In quantum many-body systems, ground states of gapped Hamiltonians have exponential decay of correlations.[2][3][4]

In quantum field theory, a continuum limit of many-body physics, a gapped Hamiltonian induces a mass gap.

References

1. "quantum mechanics - What does it mean for a Hamiltonian or system to be gapped or gapless?". Physics Stack Exchange. Retrieved 2019-02-02.
2. 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.
3. 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.
4. 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.