The Functional Integral formulation of the Schrieffer-Wolff transformation

TL;DR

This paper develops a path integral formulation of the Schrieffer-Wolff transformation, providing a systematic and symmetry-aware method to derive effective low-energy models from the Anderson model, including geometric phase considerations.

Contribution

It introduces a novel path integral approach to the Schrieffer-Wolff transformation, enabling systematic derivation of low-energy models with symmetry insights and geometric phases.

Findings

Path integral formulation of the Schrieffer-Wolff transformation.

Inclusion of Berry phase in the effective spin path integral.

Facilitates generalization to more complex models.

Abstract

We revisit the Schrieffer-Wolff transformation and present a path integral version of this important canonical transformation. The equivalence between the low-energy sector of the Anderson model in the so-called local moment regime and the spin-isotropic Kondo model is usually established via a canonical transformation performed on the Hamiltonian, followed by a projection. Here we present a path integral formulation of the Schrieffer-Wolff transformation which relates the functional integral form of the partition function of the Anderson model to that of its effective low-energy model. The resulting functional integral assumes the form of a spin path integral and includes a geometric phase factor, i.e. a Berry phase. Our approach stresses the underlying symmetries of the model and allows for a straightforward generalization of the transformation to more involved models. It thus not only sheds new light on a classic problem, it also offers a systematic route of obtaining effective low-energy models and higher order corrections.