2 8 D ec 2 00 8 1 Schwinger Bosons Approaches to Quantum Antiferromagnetism

The use of large N approximations to treat strongly interacting quantum systems been very extensive in the last decade. The approach originated in elementary particles theory, but has found many applications in condensed matter physics. Initially, the large N expansion was developed for the Kondo and Anderson models of magnetic impurities in metals. Soon thereafter it was extended to the Kondo and Anderson lattice models for mixed valence and heavy fermions phenomena in rare earth compounds [1, 2]. In these notes we shall formulate and apply the large N approach to the quantum Heisenberg model [3–6]. This method provides an additional avenue to the static and dynamical correlations of quantum magnets. The mean field theories derived below can describe both ordered and disordered phases, at zero and at finite temperatures, and they complement the semiclassical approaches. Generally speaking, the parameter N labels an internal SU(N) symmetry at each lattice site (i.e., the number of " flavors " a Schwinger boson or a constrained fermion can have). In most cases, the large N approximation has been applied to treat spin Hamiltonians, where the symmetry is SU(2), and N is therefore not a truly large parameter. Nevertheless, the 1/N expansion provides an easy method for obtaining simple mean field theories. These have been found to be surprisingly successful as well. The large N approach handles strong local interactions in terms of constraints. It is not a perturbative expansion in the size of the interactions but rather a saddle point expansion which usually preserves the spin symmetry of the Hamiltonian. The Hamiltonians are written as a sum of terms O † ij O ij , which are biquadratic in the Schwinger boson creation and annihilation opera