By Professor Bruno Nachtergaele, Professor Jan Philip Solovej, Professor Jakob Yngvason (auth.), Professor Bruno Nachtergaele, Professor Jan Philip Solovej, Professor Jakob Yngvason (eds.)

The first a part of this booklet comprises E. Lieb's basic contributions to the mathematical conception of Condensed subject Physics. frequently thought of the founder of the sector, E. Lieb demonstrates his skill to choose crucial matters and to formulate them as well-defined mathematical difficulties and, ultimately, to unravel them. the second one half offers Lieb's paintings on integrable types. His groundbreaking articles helped to set up precisely Soluble versions as a flourishing examine box in its personal correct. The papers gathered during this quantity have additionally been rigorously annotated via the editors.

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The shortest proof is to return to H and the SZ -0 subspace (in the original variables). For each U > 0 the ground state "'(U) is nondegenerate as has been shown and I want to prove that 2S -I B 1 - 1A I. The nondegeneracy of the ground state for all U> 0 implies that the S of this unique ground state must be independent of U, for otherwise continuity in U would imply a degeneracy for some value of U > O. However, when U is very large, H, as stated before, is equivalent (to leading order in U) to h, the Heisenberg antiferromagnetic Hamiltonian defined in remark (4).

2. LYKos, P. G. & R. G. PARR. 1956. J. Chern. Phys. 24: 1166; LYKos, P. G. & R. G. Parr. 1956. J. Chern. Phys. 25: 1301. 3. HEITLER, W. & F. LoNDON. 1927. Z. Phys. 44: 455. 4. MATTHEISS, L. F. 1961. Phys. Rev. 123: 1209. 5. HUBBARD, J. 1963. Proc. Roy. Soc. (London) A276: 238. 6. LINDERBERG, J. & Y. {)HRN. 1965. Proc. Roy. Soc. (London) A285: 445. 7. LIEB, E. H. & F. Y. Wu. 1968. Phys. Rev. Lett. 20: 1445. 8. {)HRN, Y. & J. LINDERBERG. 1965. Phys. Rev. 139: AI063. 9. KEARNS, D. R. 1962. J. Chem.

1970. J. Math. Phys. 11: 3317. 19425 54 Phys. Rev. Lett. 62, 1201-1204 (1989) PHYSICAL REVIEW LETTERS VOLUME 62, NUMBER 10 6 MARCH 1989 Two Theorems on the Hubbard Model Elliott H. Lieb Departments of Physics and Mathematics. Princeton University. P. 0. Box 708. Princeton. New Jersey 08544 (Received 12 December 1988) In the attractive Hubbard model (and some extended versions of it), the ground state is proved to have spin angular momentum S =0 for every (even) electron filling. In the repulsive case, and with a bipartite lattice and a half-filled band, the ground state has S - t II B I - I A II, where IB I (I A I) is the number of sites in the B (A) sublattice.