Ordering Energy Levels of Interacting Spin Systems
Abstract
The total spin S is a good quantum number in problems of interacting spins. We have shown that for rather general antiferromagnetic or ferrimagnetic Hamiltonians, which need not exhibit translational invariance, the lowest energy eigenvalue for each value of S [denoted E(S) ] is ordered in a natural way. In antiferromagnetism, E(S + 1) > E(S) for S ≥ 0. In ferrimagnetism, E(S + 1) > E(S) for S≥S, and in addition the ground state belongs to S≤S. S is defined as follows: Let the maximum spin of the A sublattice be SA and of the B sublattice SB; then S≡|SA−SB|. Antiferromagnetism is treated as the special case of S=0. We also briefly discuss the structure of the lowest eigenfunctions in an external magnetic field.
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Based on Marshall's theorem and its extension, the absolute ground state has total spin zero for N spins on a bipartite lattice with nearest-neighbor antiferromagnetic Heisenberg-type interactions.
“Based on Marshall's theorem[35] and its extension[36] the absolute ground state has total spin zero ( S 2 = 0 ) for N spins on a bipartite lattice with nearest-neighbor antiferromagnetic Heisenberg-type interactions.”
From Quantum simulation of the wavefunction to probe frustrated Heisenberg spin systems (2011) · cites this paper as [36]
Cites
Two soluble models of an antiferromagnetic chain
Lieb, É. H., Schultz, T. D., & Mattis, D. C. (1961) · Annals of Physics · doi:10.1016/0003-4916(61)90115-4 · Cited by 4,292 · see citing works
Self-citationTheory of Ferromagnetism and the Ordering of Electronic Energy Levels
Lieb, É. H., & Mattis, D. C. (1962) · Physical Review · doi:10.1103/physrev.125.164 · Cited by 618 · see citing works
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Finitely correlated states on quantum spin chains
Fannes, M., Nachtergaele, B., & Werner, R. F. (1992) · Communications in Mathematical Physics · doi:10.1007/bf02099178 · Cited by 1,422 · see citing works
Theory of Quantum Annealing of an Ising Spin Glass
Santoro, G. E., Martoňák, R., Tosatti, E., & Car, R. (2002) · Science · doi:10.1126/science.1068774 · Cited by 734 · see citing works
Quantum simulation of the wavefunction to probe frustrated Heisenberg spin systems
Ma, X.-S., Dakic, B., Naylor, W., Zeilinger, A., & Walther, P. (2011) · Nature Physics · doi:10.1038/nphys1919 · Cited by 179 · see citing works
- Authors
- Lieb, E., & Mattis, D.
- Affiliations
- Thomas J. Watson Research Center, International Business Machines Corporation, Yorktown Heights, New York
- IBM Research - Thomas J. Watson Research Center (US)
- Venue
- Journal of Mathematical Physics
- Publisher
- AIP Publishing
- Published
- 1962-07-01
- Volume
- 3
- Issue
- 4
- Pages
- 749-751
- ISSN
- 0022-2488 (ISSN-L)
- 1089-7658
- OpenAlex
- W2146325398
- Cited by
- 882
- References
- 2
- Citations by year
- 2026: 15 · 2025: 27 · 2024: 16 · 2023: 15 · 2022: 17 · 2021: 34 · 2020: 46 · 2019: 47 · 2018: 27 · 2017: 23 · 2016: 23 · 2015: 21 · 2014: 22 · 2013: 16 · 2012: 25
- FWCI
- 0.42
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Journal of Mathematical Physics (publishedVersion)
- Topics
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Advanced Condensed Matter Physics
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Theoretical and Computational Physics
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Physics of Superconductivity and Magnetism
- Keywords
- interacting spin systems, ground state energy, antiferromagnetism, ferrimagnetism, external magnetic field
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