Introduction.- Basics of quantum spin systems.-Long-range order and spontaneous symmetry breaking in the classical and quantum Ising models.- Long-range order and spontaneous symmetry breaking in the antiferromagnetic Heisenberg model.- Long-range order and "spontaneous symmetry breaking" in Bose-Einstein condensates.-Affleck-Kennedy-Lieb-Tasaki model.- Haldane phase.-The origin of ferromagnetism.- Mathematical appendices.- Solutions.- Index.

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