Abstract
We explore the formation and collective modes of Bose-Einstein condensate of Dirac magnons (Dirac BEC). While we focus on two-dimensional Dirac magnons, an employed approach is general and could be used to describe Bose-Einstein condensates with linear quasiparticle spectrum in various systems. By using a phenomenological multicomponent model of pumped boson population together with bosons residing at Dirac nodes, the formation and time evolution of condensates of Dirac bosons is investigated. The condensate coherence and its multicomponent nature are manifested in the Rabi oscillations whose period is determined by the gap in the spin-wave spectrum. A Dirac nature of the condensates could be also probed by the spectrum of collective modes. It is shown that the Haldane gap provides an efficient means to tune between the gapped and gapless collective modes as well as controls their stability.
Highlights
Discrete parity and time-reversal symmetries as well as non-Bravais lattices lead to relativisticlike Dirac energy spectrum in what is called fermionic Dirac and Weyl semimetals [1,2,3,4,5,6,7,8,9,10,11,12,13]
Since the pumping scheme and a few other details depend on the nature of Bose-Einstein condensates and, in general, are different for polaritons, magnons, Cooper pairs, etc., we focus on the case of 2D Dirac magnons. (For pumping and dynamical instabilities in fermionic Dirac matter, see, e.g., Ref. [49].) As we discussed in the Introduction, Dirac magnons can be realized in, e.g., CrI3
In the case of relatively large wave vectors, different techniques might be required to resolve magnons belonging to different Dirac nodes or sublattices
Summary
Discrete parity and time-reversal symmetries as well as non-Bravais lattices lead to relativisticlike Dirac energy spectrum in what is called fermionic Dirac and Weyl semimetals [1,2,3,4,5,6,7,8,9,10,11,12,13]. The bosonic nature of excitations and pumping allow for an accumulation of Dirac bosons in the state with the same energy and opens up the possibility of Bose-Einstein condensation at the Dirac point This uncovers a treasure trove of various nontrivial effects discussed for conventional BECs [50,51,52]. By varying the free energy (1) with respect to the fields ψa∗,ζ and ψb∗,ζ , we derive the following Gross-Pitaevskii equations for Dirac BECs at A and B sublattices: i∂t ψa,ζ = (c0 − μ + )ψa,ζ − iv[(ζ ∂x − i∂y )ψb,ζ ] These are nonlinear Dirac equations where bosonic fields have four degrees of freedom: two pseudospins and two valley or chirality indices. As we discuss Dirac Bose-Einstein condensation can be achieved in a steady state created by pumping
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