Dirac Hamiltonians for bosonic spectra

Abstract

Dirac materials are of great interest as condensed matter realizations of the Dirac and Weyl equations. In particular, they serve as a starting point for the study of topological phases. This physics has been extensively studied in electronic systems such as graphene, Weyl, and Dirac semimetals. In contrast, recent studies have highlighted several examples of Dirac-like cones in collective excitation spectra, viz. in phonon, magnon, and triplon bands. These cannot be directly related to the Dirac or Weyl equations as they are bosonic in nature with pseudounitary band bases. We address this issue by constructing a generic deformation scheme that maps any fermionic Hamiltonian and its spectrum to that of a bosonic problem. In particular, we show that any Dirac-like equation can be deformed into a suitable bosonic form. The resulting bosonic spectra bear a two-to-one relation to that of the parent Dirac system. Their dispersions inherit several interesting properties including conical band touching points and a gap-opening role for 'mass' terms. The relationship also extends to the band eigenvectors with the bosonic states carrying the same Berry connections as the parent fermionic states. The bosonic bands thus inherit topological character as well. If the parent fermionic system has nontrivial topology that leads to midgap surface states, the bosonic analog also hosts surface states that lie within the corresponding band gap. The proposed bosonic Dirac structure appears in several known models. In materials, it is realized in Ba 2 CuSi 2 O 6 Cl 2 and possibly in CoTiO 3 as well as in paramagnetic honeycomb ruthenates. Our results allow for a rigorous understanding of Dirac phononic and magnonic systems and enable concrete predictions, e.g., of surface states in magnonic topological insulators and Weyl semimetals.

Topological Magnons: A Review

“…This includes the generalization of Dirac crossings to three dimensions where they are four-fold degenerate (57). The presence of touching points in reasonable models has been a motivating factor in exploring some of their detailed consequences such as the effects of interactions, strain and electric field gradients (52,58,59,60,61,62).…”

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

“…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.…”

Fragility of Z2 topological invariant characterizing triplet excitations in a bilayer kagome magnet

“…Such excitations are bosonic, and acquire Berry phases as a consequence of spin-orbit coupling, usually in the form of Dzyaloshniski-Moriya interactions [17]. As a result, both magnon [18][19][20][21][22][23][24][25][26][27][28][29][30] and triplon [31][32][33] bands can exhibit nontrivial Chern indices, in direct analog with TI's [34]. These systems exhibit exactly the same topologically protected edge modes as their electronic counterparts [18,20,25,31], and can be indexed in the same way, even in the presence of disorder [35,36].…”