This is a textbook on quantum magnetism and on modern techniques in strongly correlated electron systems. It explains concepts and techniques that are useful in widely diverse areas such as quantum spin systems, the fractional quantum Hall effect, superfluidity, and high-temperature superconductivity. The book starts with interactions between electrons and spin exchange; the Heisenberg and t-J models are derived from the Hubbard model, and itinerant magnetism is presented as a variational theory. Superconductivity and charge ordering in the negative-U Hubbard model are described by pseudospins. The Heisenberg model is used to elucidate general concepts such as spontaneous symmetry breaking, Goldstone modes and spin waves, thermal and quantum disorder in low dimensions, Haldane's gap. The text emphasizes nonperturbative approaches: ground state theorems, variational wavefunctions, parent Hamiltonians, single mode excitations, the spin path integral and its large-S expansion, Haldane's mapping, Polyakov's weak-coupling renormalization, large-N expansions for constrained systems, and the semiclassical theory of the t-J model. Mathematical derivations are self contained. Appendices provide standard many-body tools including second quantization, Grassmann variables, generating functionals, linear response, correlation functions, Fermi and Bose coherent-states path integrals, Matsubara representation, and the method of steepest descents. There are guided bibliographies and exercises at the end of chapters.

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