  The Schrieffer-Wolff (SW) method is a version of degenerate perturbation
theory in which the low-energy effective Hamiltonian H_{eff} is obtained from
the exact Hamiltonian by a unitary transformation decoupling the low-energy and
high-energy subspaces. We give a self-contained summary of the SW method with a
focus on rigorous results. We begin with an exact definition of the SW
transformation in terms of the so-called direct rotation between linear
subspaces. From this we obtain elementary proofs of several important
properties of H_{eff} such as the linked cluster theorem. We then study the
perturbative version of the SW transformation obtained from a Taylor series
representation of the direct rotation. Our perturbative approach provides a
systematic diagram technique for computing high-order corrections to H_{eff}.
We then specialize the SW method to quantum spin lattices with short-range
interactions. We establish unitary equivalence between effective low-energy
Hamiltonians obtained using two different versions of the SW method studied in
the literature. Finally, we derive an upper bound on the precision up to which
the ground state energy of the n-th order effective Hamiltonian approximates
the exact ground state energy.
