Krein-unitary Schrieffer-Wolff transformation and band touchings in bosonic Bogoliubov-de Gennes and other Krein-Hermitian Hamiltonians

TL;DR

This paper develops a Krein-unitary Schrieffer-Wolff transformation for Krein-Hermitian Hamiltonians, enabling effective subspace analysis and providing insights into band touchings and topological features in bosonic BdG systems.

Contribution

It introduces a perturbative Krein-unitary Schrieffer-Wolff transformation applicable to finite-size Krein-Hermitian Hamiltonians, facilitating analysis of band touchings and topological properties.

Findings

Effective Hamiltonian preserves Krein Hermiticity and stability.

Justifies codimension-based analysis of band touchings.

Revisits topological magnon band touchings in materials.

Abstract

Krein-Hermitian Hamiltonians, i.e., Hamiltonians Hermitian with respect to an indefinite inner product, have emerged as an important class of non-Hermitian Hamiltonians in physics, encompassing both single-particle bosonic Bogoliubov-de Gennes (BdG) Hamiltonians and so-called "$PT$-symmetric" non-Hermitian Hamiltonians. In particular, they have attracted considerable scrutiny owing to the recent surge in interest for boson topology. Motivated by these developments, we formulate a perturbative Krein-unitary Schrieffer-Wolff transformation for finite-size dynamically stable Krein-Hermitian Hamiltonians, yielding an effective Hamiltonian for a subspace of interest. The effective Hamiltonian is Krein Hermitian and, for sufficiently small perturbations, also dynamically stable. As an application, we use this transformation to justify codimension-based analyses of band touchings in bosonic BdG Hamiltonians, which complement topological characterization. We use this simple approach based on symmetry and codimension to revisit known topological magnon band touchings in several materials of recent interest.