Squaring the fermion: The threefold way and the fate of zero modes

Krein-Hermitian Schrieffer-Wolff transformation and band touchings in bosonic BdG Hamiltonians

“…The Hamiltonian Ĥ is thermodynamically stable, meaning it is bounded below, if and only if H k ≥ 0 at all k [11]. In a linear spin wave system, for example, H k ≥ 0 holds as long as the underlying classical ground state is a local minimum of the classical energy.…”

“…In a linear spin wave system, for example, H k ≥ 0 holds as long as the underlying classical ground state is a local minimum of the classical energy. In many physical systems for which H k is positive semidefinite but not positive definite, the zero eigenspace corresponds to the Goldstone mode; in such systems, H k > 0 for k = 0 [3,11].…”

“…Non-Hermitian Hamiltonians have become a central field of study in condensed-matter physics in the last decades [1]. Important examples of systems governed by non-Hermitian Hamiltonians are quadratic bosonic Bogoliubov-de Gennes (BdG) Hamiltonians [2,3], which have been the subject of renewed interest recently in the context of topological systems [4][5][6][7][8][9][10][11][12][13], and so-called "P T -symmetric" non-Hermitian Hamiltonians, which initially attracted interest because they could give rise to real spectra despite not being Hermitian [14,15]. Note that in the context of non-Hermitian Hamiltonians, "P T symmetry" has come to refer to any antilinear symmetry of a specific form [10], and not necessarily to space-time inversion symmetry.…”

“…Note that in the context of non-Hermitian Hamiltonians, "P T symmetry" has come to refer to any antilinear symmetry of a specific form [10], and not necessarily to space-time inversion symmetry. Both these examples fall into a class of non-Hermitian Hamiltonians called Krein-Hermitian [10,[16][17][18][19][20][21][22][23], also referred to in the literature as "pseudo-Hermitian" [10,11,[16][17][18][19]24] or "para-Hermitian" [25].…”

“…They have spectral properties that enable them to describe stable physical systems: the phase diagram of a Krein-Hermitian Hamiltonian typically includes dynamically stable regions, in which the Hamiltonian is diagonalizable with real eigenvalues and with eigenvectors that are orthonormal with respect to the indefinite inner product. Lately, the mathematics of Krein-Hermitian Hamiltonians and Krein spaces more generally have seen growing interest and adoption among theoretical physicists as a tool to approach these specific non-Hermitian problems [10,11,22,25].…”