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

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.

Updated to published version. Slight change to title. Some additional explanations and figures were added based on referee comments. 21 pages, 5 figures

References in corpus (17)

• Making Sense of Non-Hermitian Hamiltonians
• Phase transition between the quantum spin Hall and insulator phases in 3D: emergence of a topological gapless phase
• k.p theory for two-dimensional transition metal dichalcogenide semiconductors
• Dissipative Phase Transition in Central Spin Systems
• Spontaneous Magnon Decays
• $PT$ Symmetric Real Dirac Fermions and Semimetals
• Phononic Helical Nodal Lines with $\mathcal{P}\mathcal{T}$ Protection in MoB${}_2$
• Interaction-stabilized topological magnon insulator in ferromagnets
• Order-by-Disorder from Bond-Dependent Exchange and Intensity Signature of Nodal Quasiparticles in a Honeycomb Cobaltate
• Spin-Space Groups and Magnon Band Topology
• XY magnetism, Kitaev exchange, and long-range frustration in the $J_{\mathrm{eff}}=1/2$ honeycomb cobaltates
• Magnon nodal-line semimetals and drumhead surface states in anisotropic pyrochlore ferromagnets
• Squaring the fermion: The threefold way and the fate of zero modes
• Krein-Space Formulation of PT-Symmetry, CPT-Inner Products, and Pseudo-Hermiticity
• PT-Symmetric Quantum Theory Defined in a Krein Space
• General Aspects of PT-Symmetric and P-Self-Adjoint Quantum Theory in a Krein Space
• Squeezing-induced Topological Gap Opening on Bosonic Bogoliubov Excitations

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