Schrieffer-Wolff transformation for quantum many-body systems

TL;DR

This paper provides a rigorous and systematic overview of the Schrieffer-Wolff transformation, including its exact and perturbative forms, with applications to quantum spin systems and bounds on energy approximation accuracy.

Contribution

It offers a self-contained, rigorous summary of the SW method, including proofs, diagram techniques, and bounds, with specialization to quantum spin lattices.

Findings

Established unitary equivalence between different SW methods.

Provided a systematic diagram technique for high-order corrections.

Derived bounds on ground state energy approximation accuracy.

Abstract

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.