The geometry of the Hermitian matrix space and the Schrieffer--Wolff transformation

TL;DR

This paper provides a geometric interpretation of the Schrieffer--Wolff transformation in quantum mechanics, relating eigenvalue deviations to distances from degeneracy submanifolds and exploring implications for Weyl points and topological properties.

Contribution

It establishes a local coordinate chart induced by the SW transformation on the space of Hermitian matrices and proves a distance theorem linking eigenvalue standard deviation to geometric distance.

Findings

Eigenvalue standard deviation equals the distance to degeneracy submanifold divided by sqrt(k).

Standard deviation and eigenvalue differences have the same order of energy splitting.

Protection of Weyl points is proved using transversality theorem.

Abstract

In quantum mechanics, the Schrieffer--Wolff (SW) transformation (also called quasi-degenerate perturbation theory) is known as an approximative method to reduce the dimension of the Hamiltonian. We present a geometric interpretation of the SW transformation: We prove that it induces a local coordinate chart in the space of Hermitian matrices near a $k$-fold degeneracy submanifold. Inspired by this result, we establish a `distance theorem': we show that the standard deviation of $k$ neighboring eigenvalues of a Hamiltonian equals the distance of this Hamiltonian from the corresponding $k$-fold degeneracy submanifold, divided by $\sqrt{k}$. Furthermore, we investigate one-parameter perturbations of a degenerate Hamiltonian, and prove that the standard deviation and the pairwise differences of the eigenvalues lead to the same order of splitting of the energy eigenvalues, which in turn is the same as the order of distancing from the degeneracy submanifold. As applications, we prove the `protection' of Weyl points using the transversality theorem, and infer geometrical properties of certain degeneracy submanifolds based on results from quantum error correction and topological order.