PHYSICAL REVIEW RESEARCH 3, 013002 (2021)

Bose-Einstein condensate of Dirac : Pumping and collective modes

P. O. Sukhachov ,1,2,* S. Banerjee ,3,† and A. V. Balatsky 4,1,‡ 1Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden 2Department of Physics, Yale University, New Haven, Connecticut 06520, USA 3Theoretical Physics III, Center for Electronic Correlations and Magnetism, Institute of Physics, University of Augsburg, 86135 Augsburg, Germany 4Department of Physics, University of Connecticut, Storrs, Connecticut 06269, USA

(Received 3 August 2020; revised 9 December 2020; accepted 10 December 2020; published 4 January 2021)

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 spectrum in various systems. By using a phenomenological multicomponent model of pumped population together with 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 -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.

DOI: 10.1103/PhysRevResearch.3.013002

I. INTRODUCTION two-dimensional (2D) honeycomb lattice [23,27]. The cor- responding Dirac materials [27] such as van der Discrete parity and time-reversal symmetries as well as Waals crystals Cr Ge Te [28] and the transition metal tri- non-Bravais lattices lead to relativisticlike Dirac energy 2 2 6 halides family AX (where A = Cr and X = F,Cl,Br,I), e.g., spectrum in what is called fermionic Dirac and Weyl semimet- 3 CrI [23–25,29] received attention recently. These materials als [1–13]. The same attributes can be arranged in the case 3 are layered ferromagnets with typical ferromagnetic transi- where the lattice is populated with bosons. Thus, one can eas- tion in the temperature range of 10–50 K. Transition metal ily extend the notion of Dirac and Weyl materials to include trihalides were known for a long time [30,31] and can be bosonic Dirac materials that host multicomponent elementary viewed as a magnetic analog of ABC stacked graphite. In three bosonic excitations described effectively by the Dirac or Weyl dimensions (3D), a magnonic Weyl state was predicted in Hamiltonian. Dirac bosons were realized in photonic [14–19] pyrochlores [32–34]. Linear crossing in the magnon spectrum and phononic [20–22] crystals and magnets [23–25]. Honey- was indeed observed in the antiferromagnet Cu TeO [35] and comb arrays of superconducting grains were also predicted 3 6 the multiferroic ferrimagnet Cu OSeO [36]. to host bosonic Dirac excitations [26]. Photonic and phononic 2 3 A wide range of properties was observed within this emer- nodal excitations are seen in specifically crafted metamaterials gent class of Dirac magnon materials. We mention interaction that reveal Dirac points when illuminated by electromag- effects [23,37] that renormalize the slope of the disper- netic or acoustic waves. For example, a linear graphenelike sion, topological properties [36,38–44], strain effects [45–47], dispersion relation of was observed in the photo- and non-Hermitian dynamics [48]. While some properties of luminescence spectrum of a honeycomb micropillar lattice in fermionic and bosonic Dirac materials are similar, there are, of Ref. [14]. course, crucial differences. The key is the difference in parti- It is known that a Dirac-type quasiparticle spec- cle statistics: there is no Pauli principle for bosons. Therefore, trum is generated for localized magnetic moments on a no exists in bosonic Dirac materials. For the excitations like magnons or polaritons, the equi- librium is zero because these excitations *[email protected] are absent in the ground state in equilibrium. Thus, no †[email protected] Bose-Einstein condensate (BEC) of magnons is possible in ‡[email protected] equilibrium. Nevertheless, one can ask if it possible to create the coherent states or even Bose-Einstein condensate of Dirac Published by the American Physical Society under the terms of the bosons and what would be the proper description? We pro- Creative Commons Attribution 4.0 International license. Further pose that the answer to this question is positive and suggest distribution of this work must maintain attribution to the author(s) a realization of coherent Dirac bosons and condensates by and the published article’s title, journal citation, and DOI. Funded creating steady-state nonequilibrium population at the Dirac by Bibsam. nodes. Such condensates could naturally be viewed as Dirac

2643-1564/2021/3(1)/013002(18) 013002-1 Published by the American Physical Society SUKHACHOV, BANERJEE, AND BALATSKY PHYSICAL REVIEW RESEARCH 3, 013002 (2021)

BECs: they would have multiple component and the nodal Dirac BEC from the magnetic Hamiltonian, and investigate spectrum described by an effective Hamiltonian resembling the time evolution of pumped condensates and the proper- that of Dirac (quasi)particles. ties of collective modes. While we focus on the case of 2D Dirac nodes are usually located at higher energies Dirac magnons, the model is general and can be applied for where, unlike , a nonvanishing population cannot be investigating various Dirac BECs. Since no Fermi surface achieved via equilibrium or gating. Therefore, one exists for bosons, we apply pumping to generate a macro- would need to produce excitations (i.e., pump the material) to scopic occupancy of magnons and ultimately a magnon BEC. populate the Dirac points, e.g., see the discussion on pumping This process is phenomenologically described by a multicom- for magnons below. (For population of the Dirac nodes via ponent system, which consists of nonlinear Gross-Pitaevskii light pulses for fermions, see Ref. [49].) equations for a Dirac magnon BEC and a rate equation for The bosonic nature of excitations and pumping allow for an a pumped magnon population. As in the case of conventional accumulation of Dirac bosons in the state with the same en- magnon BECs, the condensate appears after reaching a certain ergy and ultimately opens up the possibility of Bose-Einstein pump power. The effective Haldane gap term, allowed by the condensation at the Dirac point. This uncovers a treasure Dirac nature of magnons, lifts the degeneracy between the trove of various nontrivial effects discussed for conventional magnon densities for different pseudospins (or sublattices). BECs [50–52]. Recently, properties of BECs with Dirac Furthermore, the gap leads to the Rabi oscillations of the points in the energy spectrum were studied in Refs. [53–59] magnon densities when the Josephson coupling terms are (see also a vast literature on the nonlinear , present. If experimentally observed, such oscillations would e.g., Refs. [60–62]). Among the most interesting features, be a definitive signature of the condensate coherence. The we notice various types of vortices and solitons [55–58]. As nontrivial nature of Dirac BEC is manifested also in the spec- plausible systems for the realization of Dirac BECs, cold trum of collective modes. In our analysis, we used the standard atoms and honeycomb optical lattices were proposed. In all Bogolyubov approach and considered a homogeneous ground of these studies, the prior equilibrium BEC was subjected to state. Interestingly, there are two possibilities for the latter: a perturbing potential to form the Dirac nodes. To the best (i) magnon densities for both pseudospins are nonzero and of our knowledge, however, the possibility and properties of (ii) one of the densities vanishes. The first state spontaneously Dirac magnon BECs were not investigated before. Obviously, breaks the rotational symmetry leading to two gapless Nambu- the ultimate proof of the existence of these condensates would Goldstone modes. On the other hand, the second ground state be given by a successful experiment. Here we lay out the is characterized by a gapped Higgs mode and a gapless mode. theoretical framework that would facilitate the realizations of Depending on the model parameters, collective modes could the Dirac BEC. The framework should be understood as a be unstable. For example, the Haldane gap can be used to ac- model suitable for Bose-Einstein condensates where bosons cess different regimes for collective modes. To the best of our can be described by a relativisticlike Dirac equation. knowledge, the investigation of Dirac bosons, in particular, Magnons represent a convenient and tested platform for Dirac magnons, is in its infancy and a Dirac magnon BEC realizing quasiparticle coherent states and BEC in solids. For remains yet to be experimentally realized. We hope that our example, magnon BEC in yttrium-iron-garnet (YIG) films findings will stimulate the corresponding experimental search was recently experimentally observed and analyzed [63–71]. and the development of the field of Dirac BECs. (While we focus on magnons, the BEC of polaritons is also The paper is organized as follows. We introduce the model possible and was observed in, e.g., Refs. [72–76].) Among and key notions in Sec. II. While we focus on the case of the most notable properties, we mention spatial coherence, magnons, the model is general and could be used to describe supercurrents, and sound modes of the condensate. In order to various Dirac BECs. Pumping and time evolution of Dirac achieve a magnon BEC, one needs to accumulate magnons in magnons are phenomenologically considered in Sec. III. Sec- the minima of dispersion relation, which is usually achieved tion IV is devoted to collective modes in the Dirac BECs. The by a certain pumping protocol. Often used is the parametric results are discussed and summarized in Sec. V. A schematic pumping technique [77]. The central process in this scheme is derivation of the free-energy ansatz used in the main text is the splitting of a photon of an external oscillating electromag- given in Appendix A. A few nontrivial phases of the Dirac netic field into two magnons with arbitrary opposite momenta BEC are summarized in Appendix B. The additional results but with the half-frequency. These magnons subsequently ac- for the population dynamics at different interaction constants cumulate at the minima of the dispersion and condense if are presented in Appendix C. Appendix E contains disper- the pumping intensity exceeds certain threshold. In contrast sion relations of collective modes for a nonzero Josephson to the case of Dirac magnons, the spin-wave spectrum in coupling. The model is extended to include the magnon con- YIG is “trivial,” i.e., it does not exhibit any Dirac crossings densate at the point and the results for the population and is well described by a conventional parabolic dispersion. dynamics are presented in Appendix D. Through this study, Moreover, the vast literature investigating the properties of we seth ¯ = kB = 1. driven BEC in YIG relies on the standard single-component models for magnon BEC, which cannot be applied to Dirac II. MODEL . For the case under consideration, we assume the nodal In this section, we present a phenomenological approach to structure of spin excitations (Dirac nodes) and aim to de- Dirac BECs. The approach is similar to that for conventional scribe the formation of long-lived and ultimately condensed BECs [52] albeit it incorporates the pseudospin and a linear magnons at Dirac nodes. We derive a minimal model for a Dirac dispersion relation. We start with the following minimal

013002-2 BOSE-EINSTEIN CONDENSATE OF DIRAC MAGNONS: … PHYSICAL REVIEW RESEARCH 3, 013002 (2021) ansatz for the free-energy density (for a schematic derivation shown schematically in Fig. 1. Here, red and brown shaded in the case of magnons, see Appendix A): regions correspond to the accumulated magnons (or other  Dirac bosonic quasiparticles) and spikes represent conden- = − μ + − − † τ ⊗ σ · ∇  F (c0 )ntot (na,ζ nb,ζ ) iv ( z ) sates. Note that unlike fermionic systems, energy is calculated ζ =± with respect to the bottom of the bands. Two Dirac points are    =± ω = ω = 1 1 situated at k kD in momentum space and at a b c0 + g n ,ζ n ,ζ + g n ,ζ n ,ζ + g n ,ζ n ,ζ 2 a a a 2 b b b ab a b in energy. As one can see from Fig. 1(b), the presence of the ζ =± term opens the gap in the spectrum. 1  By varying the free energy (1) with respect to the fields ∗ ∗ ∗ ∗ ∗ ∗ ∗ + (uabψ ,ζ ψ ,ζ ψb,ζ ψb,ζ + u ψ ,ζ ψ ,ζ ψa,ζ ψa,ζ ). ψ ψ 2 a a ab b b a,ζ and b,ζ , we derive the following Gross-Pitaevskii equa- ζ =± tions for Dirac BECs at A and B sublattices: (1) i∂t ψa,ζ = (c0 − μ + )ψa,ζ − iv[(ζ∂x − i∂y )ψb,ζ ] The four-component of Dirac BEC is + ψ + ψ + ψ∗ ψ 2, gana,ζ a,ζ gabnb,ζ a,ζ uab a,ζ ( b,ζ ) (3) T  = (ψa,+,ψb,+,ψb,−,ψa,−) , (2) i∂t ψb,ζ = (c0 − μ − )ψb,ζ − iv[(ζ∂x + i∂y )ψa,ζ ] where ζ =±denotes the K or K valley in 2D or chirality + ψ + ψ + ∗ ψ∗ ψ 2. in 3D, ψa,ζ and ψb,ζ are the condensate wave functions for gbnb,ζ b,ζ gabna,ζ b,ζ uab b,ζ ( a,ζ ) (4) the different pseudospins, e.g., A and B sublattices for 2D These are nonlinear Dirac equations where bosonic fields have σ hexagonal lattices, is the vector of the in four degrees of freedom: two pseudospins and two valley or τ the pseudospin space, z is the Pauli in the valley chirality indices. = ψ∗ ψ = , space, ni,ζ i,ζ i,ζ with i a b is the density for a certain It is interesting to notice the nontrivial topology of the = + pseudospin, ntot ζ =± (na,ζ nb,ζ ) is the total condensate multicomponent BEC. If all coupling terms are ignored, density, c is the position of the Dirac nodes in energy, and μ 0 the corresponding state would correspond to the U(1)a+ × is the effective chemical potential of condensates. Finally, the U(1) − × U(1) + × U(1) − symmetry. Therefore, one would a b b term quantified by denotes the Haldane gap in 2D. (Note expect the rich phase diagram with multiple phases that could ∝ that the term plays a different role in 3D where it shifts emerge in Dirac BECs, where relative phases of these sym- the Weyl nodes of opposite chiralities in momentum space and metries can be broken. For simplicity and to make an explicit is called the chiral shift [78]. In what follows, however, we case, we will assume that relative oscillations of the conden- focus only on the 2D case.) Furthermore, we notice that the sates in both valleys are suppressed and there are only two part of the free energy linear in the density is similar to that for flavor symmetries, which are explicitly broken to U(1)a × other Dirac (quasi)particles. In particular, the part with spatial U(1)b. In other words, we consider a symmetric response derivatives resembles that for Dirac fermions. Moreover, there of the valleys. Therefore, we fix ζ =+and omit the valley are two pseudospin or sublattice degrees of freedom. The fact index when it is not necessary. Nevertheless, we mention some that kinetic energy terms and Hilbert space of the excitations nontrivial phases of the noninteracting BECs in Appendix B. are very similar to the case of known Dirac excitations allows Unlike fermions, where there is a Fermi surface that can us to dub the condensate as the Dirac BEC. be tuned to intersect Dirac points, bosons occupy the lowest- In addition to the conventional for Dirac systems lin- energy state. Therefore, in order to investigate the properties ear terms, we included also the nonlinear interaction terms of the Dirac BEC, one first needs to achieve a sufficient quasi- quantified by constants ga, gb, and gab. While the former particle density at the Dirac points. As we discuss in the next two correspond to the interaction strength of boson densities section, Dirac Bose-Einstein condensation can be achieved in of the same pseudospin, the latter describes the interpseu- a steady state created by pumping. dospin interaction. In addition, the Josephson coupling term quantified by u is considered. This term is reminiscent ab III. GENERATION OF DIRAC BOSE-EINSTEIN to the spin-orbital coupling term in conventional spin-1 CONDENSATES VIA PUMPING BECs [79]. Cross-condensate Josephson terms describe the internal “Josephson effect” where a pair of magnons of type In this section, we consider pumping and time evolution of a converts into a pair of magnons of type b. As we will Dirac BECs. Since the pumping scheme and a few other de- show below, the Josephson coupling term mixes the phases tails depend on the nature of Bose-Einstein condensates and, of the condensate wave functions corresponding to different in general, are different for polaritons, magnons, Cooper pairs, pseudospins and might significantly affect the dynamics of the etc., we focus on the case of 2D Dirac magnons. (For pumping condensates. and dynamical instabilities in fermionic Dirac matter, see, Finally, we notice that several other terms are possible in e.g., Ref. [49].) As we discussed in the Introduction, Dirac the free energy, e.g., terms corresponding to the intervalley magnons can be realized in, e.g., CrI3 (see Refs. [23–25]). mixing ∝na,−ζ na,ζ . For the sake of simplicity, we consider the Note that the Haldane gap for Dirac magnons can be generated effect of only a few parameters on Dirac condensates. A more via the pseudo-Zeeman effect (for a discussion of the strain- detailed discussion is left for the future investigation. induced pseudo-Zeeman effect for fermions in , see The 2D Dirac dispersion relation (see Appendix A for the Ref. [80]). In addition, the relatively large gap in the Dirac discussion of the Dirac spin-wave dispersion relation) and spectrum could appear due to the Kitaev interaction. For a BECs corresponding to different valleys and pseudospins are recent study in CrI3, see, e.g., Ref. [25].

013002-3 SUKHACHOV, BANERJEE, AND BALATSKY PHYSICAL REVIEW RESEARCH 3, 013002 (2021)

(a) (b)

FIG. 1. Schematic representation of the Dirac spectrum and BECs. The condensates of Dirac magnons occur at the Dirac points located at k =±kD in momentum space. (a), (b) Correspond to gapless = 0 and gapped = 0 spectra, respectively. Each of the condensates contains two components related to the pseudospin (sublattice) degree of freedom. The corresponding densities are denoted as na,ζ and nb,ζ with ζ =±. The density n3 corresponds to the pumped population of bosons with energy ω3 (see Sec. III for the description of the pumping mechanism). Finally, n0 denotes the population at the point (see also Appendix D).

A. Three-component model 3 denotes the decay rate. Finally, the term 3P˜(t ) describes To describe the pumped Dirac BEC, we introduce a phe- the direct inflow of magnons caused by the pump. (Note nomenological three-component model (or five-component if that explicit dynamics of noncondensed magnons [81]atthe the valley index is taken into account). In addition to the Dirac nodes is not included in the above system. In the case magnon populations at the A and B sublattices for the K under consideration, it is implicitly taken into account in the point, we include the pumped magnon population with the inflow terms.) Therefore, the formation of the condensates is energy ω3 (see also Fig. 1). The extension of this model to the a multistep process where the pumping first creates a magnon case where the magnon BEC is formed at the point is pre- gas which decays into the Dirac magnons. sented in Appendix D. These populations could be achieved For the sake of simplicity, we consider a simple steplike by the parametric pumping similarly to the case of conven- profile for the pump power √ tional magnetic materials [77] or any other suitable pumping P˜(t ) = Pmax θ(t − tBegin )θ(tEnd − t ). (8) scheme. The results below do not depend on the details of the pumping. The system reads as Here the pump starts at t = tBegin and ends at t = tEnd. Further, Pmax is the power of the pumping field and θ(x) is the step i∂t ψa + iaψa = (c0 − μ + )ψa − iv[(∂x − i∂y )ψb] function. The square-root dependence on the pump power is in agreement with the experimental data for YIG [66,82](see + ψ + ψ + ψ∗ ψ 2 gana a gabnb a uab a ( b) also Refs. [83,84] for the theoretical discussion). While only a + iPa3an3ψa, (5) simple steplike profile in Eq. (8) is considered in this work, the time profiles of magnon BECs can be modulated by applying i∂t ψb + ibψb = (c0 − μ − )ψb − iv[(∂x + i∂y )ψa] more complicated time-dependent pump. ∗ ∗ 2 + gbnbψb + gabnaψb + u ψ (ψa ) ab b B. Time evolution of Dirac magnons + iPb3bn3ψb, (6)  In this section, we present the numerical results for the pumped magnon populations and discuss the role of the Hal- ∂t n3 + 3n3 = 3P˜(t ) − (P3aana,ζ + P3bbnb,ζ )n3. ζ =± dane gap and the Josephson coupling terms. Unless otherwise (7) stated, we use the following set of the model parameters: = = = 0.25 t −1, c = t −1,μ= 0,= 0, Compared to Eqs. (3) and (4), we have added the dissipation a b 3 0 0 0 ψ ψ = = −1, = = , terms i a a and i b b that correspond to the magnon- ga gb t0 gab uab 0 (9) interactions. The terms with Pa3 and Pb3 describe to the inflow of magnons from the pumped population and act as the source P3a = P3b = Pa3 = Pb3 = 1, tBegin = 10 t0, tEnd = 200 t0. terms for Dirac magnons. On the other hand, the terms with (10) P3a and P3b correspond to the depletion of the pumped magnon population. The latter is described by the rate equation (7)for Here, t0 is the characteristic interaction timescale. Since no the magnon density n3. As in the case of the magnon BEC, experimental data are present for the Dirac magnon BEC,

013002-4 BOSE-EINSTEIN CONDENSATE OF DIRAC MAGNONS: … PHYSICAL REVIEW RESEARCH 3, 013002 (2021)

FIG. 2. The dependence of the magnon densities on time t at a few values of the Haldane gap for Pmax = 0.5 P0. The densities are normalized to the total magnon density at t = tEnd. Here the values of parameters given in Eqs. (9)and(10) are used. In addition, P0 is the = −2 characteristic strength of the pump (P0 t0 ). The Haldane gap leads to the splitting of the magnon densities corresponding to different pseudospins.

we use model parameters. In addition, we note that, in gen- pumped magnon population n3 quickly reaches its maximum eral, Pi3 = P3i with i = a, b. For example, a strong disparity value determined by the interplay of the pumping power and between the scattering efficiency of gaseous population to the decay rate. Magnon populations at the Dirac points occur magnons at the bottom of the energy dispersion and vice versa later in time as well as require pump power to exceed a certain is observed for YIG in Ref. [69]. Further, because of the space threshold (see Fig. 3). It is interesting to note that the densities separation between the lattice sites, the intersublattice inter- of the magnon BECs reach their maximum values at the onset action constant gab is expected to be much weaker than the of the profile. In addition, since we chose equal inflow and intrasublattice ones ga and gb. Therefore, we focus on the case outflow terms in Eqs. (5) through (7), there is a noticeable |gab|ga, gb. The results for nonvanishing gab are presented depletion of the pumped population. After turning off the in Appendix C. Qualitatively, however, they are similar to pump, both condensates and pumped population decay during the case |gab|ga, gb. Finally, we note that the interaction the time τ ∼ 1/. It is notable, also, that the total number of constants ga, gb, and gab are irrelevant for the condensate magnons is not conserved and shows a spike when the BECs dynamics in the absence of the phase-mixing terms [e.g., the start to form. This could be qualitatively explained by the fact terms with uab in Eqs. (5) and (6)]. This follows from the fact that the Dirac points accumulate more magnons during the that the density dynamics is determined by the imaginary part rise phase than they can support in the steady state. These of the Gross-Pitaevskii equations (5) and (6). magnons are released during the equilibration phase. The time profiles of magnon densities are shown in Figs. 2 Further, we investigate the effect of the Haldane gap . and 3 for several values of the pump power. As one can see, the The corresponding results are shown in Fig. 2 at a few values

FIG. 3. The dependence of the magnon densities [Dirac BEC in (a) and pumped population in (b)] on time t and pump power Pmax. Here = −2 the values of parameters given in Eqs. (9)and(10) are used. In addition, P0 is the characteristic strength of the pump (P0 t0 ). The pumped population demonstrates a box-shaped profile. On the other hand, the Dirac BEC forms much faster at larger powers.

013002-5 SUKHACHOV, BANERJEE, AND BALATSKY PHYSICAL REVIEW RESEARCH 3, 013002 (2021)

= . = . −1 FIG. 4. The dependence of the magnon densities on time t at a few values of uab for Pmax 0 5 P0 and 0 1 t0 . The densities are normalized to the total magnon density at t = tEnd. Here the values of parameters given in Eqs. (9)and(10) are used. In addition, P0 is the = −2 / > . −1 characteristic strength of the pump (P0 t0 ). The Rabi oscillations, whose period is determined by 1 at small uab and large ( 0 1 t0 = . −1 at uab 0 2 t0 ), are clearly visible for the magnon BECs. of the gap for Pmax = 0.5 P0. As one can see, the gap leads to pure states correspond to the surface of the Bloch sphere. The different densities of magnon BECs at the A and B sublattices. animation of dynamics of the Dirac magnon populations is Therefore, it can be used to effectively control the pseudospin given in the Supplemental Material [87]. The snapshot after distribution of Dirac magnon BECs. the pump is turned off (t = 300 t0) is shown in Fig. 6. Finally, let us study the effect of the Josephson coupling term uab on the time evolution of magnon densities. The cor- responding results for a few values of uab are shown in Fig. 4. IV. COLLECTIVE MODES As one can see, the Josephson coupling term ∝u leads to ab In the previous section, we considered pumping of uni- the oscillations of the Dirac BEC. Oscillations of n and n a b formly distributed magnons. To provide a deeper insight into always occur in the antiphase and the pumped magnon popu- the Dirac nature of magnon condensate and uncover properties lation remains constant. The observed oscillations are, in fact, related to the linear spectrum, one should consider coordinate- the Rabi oscillations between magnon BECs with different dependent probes. Collective modes are, perhaps, among the pseudospins. To elucidate the nature of the intercondensate most well-known, powerful, and versatile of them. oscillation and to show that they have the same scaling as the Rabi oscillation, we plot the corresponding period as a function of in Fig. 5. The period indeed scales as 1/ for large values of . The case of small is more complicated, however. For example, the period becomes a nonmonotonic function and depends, e.g., on the value of uab and other inter- action parameters. A nontrivial dynamics becomes manifested  −1 at t0 . Thus, the gap not only induces different magnon densities at the A and B sublattices, it can activate the Rabi oscillations between the condensates. We believe that the experimental observation of the Rabi oscillation of the magnon densities could serve as a definitive proof of their coherence. In the systems with small distance between the Dirac nodes in the momentum space, such observations could be, in principle, done via the Brillouin light scattering technique used for con- ventional magnons in YIG. In the case of relatively large wave vectors, different techniques might be required to resolve magnons belonging to different Dirac nodes or sublattices. Among them, we mention time-resolved magneto-optic Kerr FIG. 5. The period of intercondensate (Rabi) oscillations is effect measurements and nitrogen-vacancy (NV) centers (see shown by the black dots. The red line represents a linear fit. For large Refs. [85,86]forareview). values of , we indeed observe the expected for the Rabi oscillations The dynamics of the Dirac magnon populations can be scaling for the period as inverse gap. On the other hand, for smaller presented by introducing the vector in the density space values of the Haldane gap, the dynamic is more complicated and does (na, nb, n3)/ntot. The tip of such vector is always confined to not follow a simple linear behavior. We see a significant scatter in  −1 the one-eighth of a unit sphere. The evolution of densities then the period due to nonlinear dynamic at t0 . In this case, other resembles the dynamics of two-level quantum system whose interaction parameters also become important.

013002-6 BOSE-EINSTEIN CONDENSATE OF DIRAC MAGNONS: … PHYSICAL REVIEW RESEARCH 3, 013002 (2021)

By using Eqs. (11) and (12), the Gross-Pitaevskii equa- tions (3) and (4) lead to

0 = (c0 − μ + )ψa,0 + ganaψa,0 + gabnbψa,0 + ψ∗ ψ 2, uab a,0( b,0 ) (13)

0 = (c0 − μ − )ψb,0 + gbnbψb,0 + gabnaψb,0 + ∗ ψ∗ ψ 2. uab b,0( a,0 ) (14) It is convenient to separate the absolute value and the phase √ θ ψ = i a of the ground-state√ wave functions, i.e., a,0 nae and iθb ψb,0 = nbe . As follows from Eqs. (13) and (14), the phase of the ground-state solutions is not fixed for uab = 0. There- fore, the state has U(1)a × U(1)b symmetry. If the Josephson FIG. 6. The trajectory of the system in the density space coupling term is nonzero, we have (n , n , n )/n is shown by the red line. Since the population den- a b 3 tot θ = 2(θ − θ ) + πl, l = 0, 1, 2, 3,... (15) sities are normalized to the total density, the dynamics is always uab a b constrained to the one-eighth of a unit sphere. When the pump is iθu where uab =|uab|e ab . Further, we find an effective chemical turned on, only the pumped population n3 contributes to the total potential that allows for a solution of the Gross-Pitaevskii density. Later, the condensates develop and the density vector starts equations. We obtain to oscillate closer to the equator. Finally, the condensates decay and the system returns to its initial state. Here, the values of parameters μ = c0 + + gana + gabnb + uabnb, (16) = . −1 given in Eqs. (9)and(10) are used. In addition, we set uab 0 1 t0 and = 0.025 t −1. For the full time dependence, see Supplemental 0 μ = c0 − + gbnb + gabna + uabna. (17) Material [87]. Note that even and odd values of l correspond to positive and negative values of uab in the equation above. The system (16) Before we proceed with more detailed analysis, let us and (17) has a solution for briefly discuss the structure of collective modes. There are two 2 + (ga − gab − uab)na condensate densities na and nb. Hence, we would expect two n = . b − − (18) sound modes where the coupling is independent of the relative gb gab uab phase. Two linear modes will remain linear with renormalized (One can fix nb and solve for na, which is physically equiv- velocities once the interaction is turned on. This is what we alent). The effective chemical potential for the density (18) = = indeed find in the regime na 0 and nb 0. However, we reads as also find the state where the condensate breaks the pseudospin 2 + (g − g − u )n symmetry and one of the components vanishes in the ground μ = c + + g n + g + u a ab ab a . 0 a a ( ab ab) − − (or, more precisely, quasiground) state, e.g., nb = 0. In that gb gab uab case, we find only one gapless mode corresponding to the (19) phase mode of the condensate. The second mode is gapped and corresponds to the Higgs or amplitude mode, where the There is also an additional solution for Eqs. (13) and (14), = populations at the A and B sublattices oscillate. where nb 0 and In order to investigate the spectrum of collective excita- μ = c + + g n . (20) tions, we employ the Bogolyubov approach [50–52]forthe 0 a a magnon BEC. In this case, one looks for the solution as We note that solution (18) exists as long as nb > 0. Otherwise, only the ground state with nb = 0 and μ given in Eq. (20)is ψ = ψ + −iμt −iωt+ik·r + ∗ iω∗t−ik·r , possible. a a,0(r) e [uae va e ] (11) Therefore, even for simple uniform and static equations ψ = ψ + −iμt −iωt+ik·r + ∗ iω∗t−ik·r . for the Dirac BEC, we find two types of solutions: one with b b,0(r) e [ube vb e ] (12) nonvanishing magnon densities na = 0 and nb = 0 and the solution that breaks the pseudospin symmetry with na = 0but Here, ψ , (r) and ψ , (r) denote ground-state solutions, a 0 b 0 n = 0. The latter solution is an example of nematic instability which are, in general, nonuniform. They satisfy the time- b similar to nematic states in superconductors [88] and independent systems of the Gross-Pitaevskii equations. The ∗ ω in Bose nematics (see, e.g., Ref. [89]). perturbations are described by ua,b and va,b terms. Finally, is the frequency and k is the wave vector of collective modes. B. Free energy In order to clarify which of the two possible solutions A. Ground state for the steady-state homogenous Gross-Pitaevskii equations, Let us start with the ground-state solutions ψa,0(r) and discussed in Sec. IV A, is the true ground state, we calculate ψb,0(r). For simplicity, we consider a uniform ground state. the free energy of the system. By using Eq. (1), we derive the

013002-7 SUKHACHOV, BANERJEE, AND BALATSKY PHYSICAL REVIEW RESEARCH 3, 013002 (2021)

FIG. 7. The phase diagram of the system for a few combinations of parameters: ga and gb (a), ga and gab (b), and ga (c), as well as and na (d). Red and blue colors correspond to the ground states with nb = 0andnb = 0, respectively. By default, the values of parameters given in Eqs. (9)and(10)areused. following expression: state. As one can see, the phase diagram is nontrivial where = =    states with both nb 0 and nb 0 are possible. In general, 1 2 2 F = = g n (g + u ) however, the ground state with nb = 0 is favored for repulsive nb 0 2 b a ab ab (gab + uab − gb) > >  interactions (ga 0 and gb 0). Furthermore, it is clear that + + − 2 − − 2 the Haldane gap parameter could be used to tune the ground 2ga(gab uab) ga 4gana 4 state [see Figs. 7(c) and 7(d)]. − n (g + u )2[n (2(g + u ) − g ) − 4] a ab ab a ab ab a − 2 2 , gagbna (21) C. Dispersion relations n n = where b given in Eq. (18) was used. In the case b 0, we In this section, we use the solutions found in Sec. IV A and have a simpler expression linearize the Gross-Pitaevskii equations (3) and (4). By using =− 2. relations (11) and (12), one obtains the following equations Fnb=0 3gana (22) for ua,b and va,b: In addition to comparing the free energies, one should check whether n remains positive for the state with n = 0. b b ωu = (c + − μ)u + v(k − ik )u + 2g n u + g n u We present the phase diagram of the system for a few a 0 a x y b a a a ab b a combinations of parameters in Fig. 7. Unless otherwise stated, + ψ ψ∗ + ψ2 + ψ ψ gab a,0 b,0ub ga a,0va gab a,0 b,0vb we use the values of parameters given in Eqs. (9) and (10). + ψ ψ∗ + ψ 2 , The state with the lower free energy is a true (quasi)ground uab[2 b,0 a,0ub ( b,0 ) va] (23)

013002-8 BOSE-EINSTEIN CONDENSATE OF DIRAC MAGNONS: … PHYSICAL REVIEW RESEARCH 3, 013002 (2021)

ζ ωub = (c0 − − μ)ub + v(kx + iky )ua + 2gbnbub + gabnaub As before, we omit the valley index assuming that the val- leys have the same dynamics. Note that complex conjugation + ψ ψ∗ + ψ2 + ψ ψ gab b,0 a,0ua gb b,0vb gab a,0 b,0va was performed in Eqs. (25) and (26). + ∗ ψ ψ∗ + ψ 2 , Equating the determinant of the system (23) through (26) uab[2 a,0 b,0ua ( a,0 ) vb] (24) to zero, the spectrum of collective modes is determined. In u = −ωva = (c0 + − μ)va − v(kx + iky )vb + 2ganava + gabnbva what follows, we focus on the case ab 0. The presence of the Josephson coupling term complicates the expressions for + ψ∗ ψ + ψ∗ 2 + ψ∗ ψ∗ gab a,0 b,0vb ga( a,0 ) ua gab a,0 b,0ub the frequencies. For example, numerical calculation suggests ∗ ∗ ∗ 2 that u enhances the imaginary parts of the frequencies in the + u [2ψ ψ , v + (ψ ) u ], (25) ab ab b,0 a 0 b b,0 a θ = θ = θ = case of the phase-locked condensates ( a b uab 0). For a more detailed discussion and a few dispersion relations −ωvb = (c0 − − μ)vb − v(kx − iky )va + 2gbnbvb + gabnavb at uab = 0, see Appendix E. + ψ ψ∗ + ψ∗ 2 + ψ∗ ψ∗ gab a,0 b,0va gb( b,0 ) ub gab a,0 b,0ua In the case of the nontrivial ground-state solution given in + ψ∗ ψ + ψ∗ 2 . Eq. (18), the frequencies are uab[2 a,0 b,0va ( a,0 ) ub] (26)

2v|k | ω = 2 2 ± y − 2 − + − . ± v k (gab gb) gab gagb [2 (ga gab)na] (27) gab − gb

Note that both branches are gapless. In addition, we fix the phase of the ground-state solutions as θa = θb = 0. Even at = 0, the dispersion relation (27) is rather interesting. For example, the mode is stable (i.e., there is no large imaginary part) for 2 − − − < = 2  2 (gab gagb)(gab ga )(gab gb) 0. In the case ga gb, the stability criterion is simple ga gab, which agrees with phase- separation criterion for conventional BECs [52]. For the ground state with nb = 0, one of the collective modes (e.g., ω+) can be gapped. The full spectrum in this case is + n g − g 2  2 2 2 [2 a( a ab)] 1 2 2 2 2 4 4 2 2 ω± = v k + ± {2v k + [2 + n (g − g )] } − 4{v k + v k g n [2 + (g − g )n ]}. 2 2 a a ab a a a ab a (28)

The gap between ω+ and ω− branches reads as gapless Nambu-Goldstone mode is guaranteed because U(1)a θ ψ → i a ψ |ω = − ω = |=| + − |. symmetry a,0 e a,0 is spontaneously broken in this +(k 0) −(k 0) 2 (gb gab)na (29) case. The other mode can be gapped. It corresponds to the Note that the gap is determined both by the Haldane gap term Higgs mode of the Dirac BEC. This mode is also known as and interaction terms. In a simple case = 0, the modes at the amplitude mode because it corresponds to the oscillations = 2  2 of the amplitude of the order parameter. On the other hand, the nb 0 are stable at ga gab. In addition, while the presence of the gapless modes for 2D materials might naively signify gapless Nambu-Goldstone mode is related to the oscillations an instability towards long-range power-law correlations, it of the phase rather than the absolute value of the BEC’s wave should not play a crucial role in experimental samples. Indeed, function. in such a case, the range of correlations is effectively limited Further, we notice that the spectrum of collective modes by the size of samples and the range of probes. might contain an imaginary part. The presence of the imagi- = = We present the spectra (27) and (28)inFigs.8 and 9, nary part at small ky for nb 0 and large k for nb 0 signifies respectively. It is notable that the rotational symmetry is the dynamical instability of the system. Interestingly, only one = spontaneously broken in the interacting BEC of magnons at of the collective modes (at least at gab 0) has a nonzero = n = 0. Indeed, this is evident from the expression in Eq. (27), imaginary part for nb 0. On the other hand, both modes b = where an explicit dependence on ky is present. It can be could have a nonvanishing imaginary part at nb 0. Further- checked that the anisotropy is set by the relative phase of more, we note that the dispersion relation of the collective ω = + − = the ground-state solutions ψ , and ψ , . For example, one modes becomes trivial, i.e., ± vk,at2 na(ga gab) a 0 b 0 = could replace ky → kx for θb − θa = π/2. It is worth noting, 0. In this case, nb 0. however, that the state nb = 0 is not always the ground state Let us discuss the dependence of the frequencies of the of the system (see the discussion in Sec. IV B). Depending on collective modes on the Haldane gap parameter .Asex- the values of the interaction constants, the state with nb = 0 pected from the analytical result (27), there is no dependence could be also realized. Furthermore, both collective modes on for k xˆ. On the other hand, the Haldane gap can be are gapless. These modes can be identified with two Nambu- used to tune the dispersion relation of the collective modes Goldstone modes related to the independent oscillations of the at k yˆ. For example, as one can see from Fig. 8, one of condensates at A and B sublattices, i.e., spontaneous break- the modes (ω+) becomes unstable. The Haldane gap can iθa iθb = down of the symmetries ψa,0 → e ψa,0 and ψb,0 → e ψb,0. be also used to stabilize collective modes for nb 0(see ω+ ω− The spectrum at nb = 0, on the other hand, contains one Fig. 9). In particular, and are stable at large values gapless and one gapped mode. Indeed, the presence of a of .

013002-9 SUKHACHOV, BANERJEE, AND BALATSKY PHYSICAL REVIEW RESEARCH 3, 013002 (2021)

FIG. 8. Frequencies of the collective modes ω+ (a) and ω− (b) for nb = 0 at a few values of the Haldane gap . Here, k0 = 1/(vt0 )and ω0 = vk0. The dispersion relation for k xˆ is trivial, i.e., ω± = vk. Shaded regions denote the imaginary part of the frequencies. Dispersion relations are gapless for all cases under consideration.

Furthermore, we identify three different phases determined that could be used to access different regimes for collective by the combination of parameters 2 + (gb − gab)na at nb = modes. 0. For <−cr, where cr =|(gb − gab)na|/2, we have one gapped mode and one gapless mode. The latter mode V. SUMMARY AND OUTLOOK has a nonzero imaginary part and is unstable. These modes merge at = cr and have a soundlike dispersion relation We investigated the properties of the Bose-Einstein con- ω± = vk. One of the modes remains gapless while the other densate of Dirac quasiparticles and outlined possible routes acquires a gap for −cr <<cr. The modes are un- to achieve this state in bosonic systems. The proposed frame- stable for sufficiently large values of momentum when the work is general and can be applied for investigating various spectral branches merge. Finally, the degeneracy is lifted physical systems with Dirac bosonic dispersion. We focus on and modes become stable at  cr. For the parameters the case of 2D Dirac magnons. To achieve the Bose-Einstein used, ω+ is gapped and ω− is gapless in this case. Thus, condensation at the Dirac points, a pumping of magnons was the Haldane gap appears as an efficient tuning parameter proposed. In particular, we introduced a phenomenological

FIG. 9. Frequencies of the collective modes ω+ (a) and ω− (b) for nb = 0 at a few values of the Haldane gap . Here, k0 = 1/(vt0 ), ω0 = vk0,andcr =|(gb − gab)na|/2. The dependence on ky is the same as for kx. Shaded regions denote the imaginary part of the frequencies. Except a single point = cr, one of the modes is gapped and the other is gapless.

013002-10 BOSE-EINSTEIN CONDENSATE OF DIRAC MAGNONS: … PHYSICAL REVIEW RESEARCH 3, 013002 (2021) model consisting of the Gross-Pitaevskii equations for the of materials for investigating the Dirac BEC is, however, yet Dirac BEC amended with a rate equation for the pumped to be identified. We thus expect that the suite of interesting magnon population. We found that if the pump power reaches questions about experimental realization, topology, hydrody- a certain threshold determined by the coupling terms between namics, edge states, and collective excitations in Dirac BECs the pumped and Dirac magnon populations, magnons can would need to be further addressed. populate the Dirac nodes. Depending on the values of the The multicomponent nature of Dirac BEC means that coupling terms, a noticeable depletion might be observed for one would have at least four-component hydrodynamics with the pumped magnon populations when the BECs form. The both normal and superfluid components for the Dirac quasi- profiles of the magnon densities are nonmonotonic. Indeed, particles. Since the condensates are coherent, an interesting they reach a peak value at the onset. Then, a plateau is problem is the interference of coexisting condensates from developed for a sufficiently large pumping power. Finally, different valleys. In a state with different phases of the both condensate and pumped magnon population decay when condensates at different valleys, one would expect spatial the pump is turned off. The multicomponent condensate co- interference effects that could be observed by using the herence is manifested in the Rabi oscillations of the Dirac Brillouin light scattering, just as for conventional magnon magnon populations allowed by the Haldane-type gap term BEC [67], as well as time-resolved magneto-optic Kerr ef- and the Josephson coupling. These features of the magnon fect measurements [86] and nitrogen-vacancy centers [85]. It populations are readily accessible by the Brillouin light scat- would be interesting also to investigate the formation of the tering technique used for conventional magnons in YIG [67]. condensates in finite samples and the effect of the correspond- Among other techniques that might be required to resolve ing edge (surface) modes. magnons belonging to different Dirac nodes or sublattices, Finally, let us discuss a few limitations of this study. The we mention time-resolved magneto-optic Kerr effect measure- employed model contains a minimal number of terms allowed ments [86] and nitrogen-vacancy centers [85]. by the symmetries of the system. Clearly, several other terms The Dirac nature of magnon BEC is manifested also in could be considered. Among them, we mention the terms the spectrum of collective modes. Even for a simple model corresponding to the intervalley interactions and nonlinear de- at hand, where only the dynamics of a single valley is consid- cay terms. While their effects could be interesting, additional ered, we found that the spectrum is rather rich. In particular, terms greatly expand the parameter space of the problem. two uniform (quasi)ground states are possible, where the Therefore, the corresponding studies will be reported else- magnon density is distributed among the pseudospin (sub- where. Further, the pumping model proposed in this study is lattice) degrees of freedom or is accumulated at a certain phenomenological. The derivation of the corresponding mi- sublattice. In the first case, the rotation symmetry is sponta- croscopic model as well as its application to realistic materials neously broken, leading to a directional dependence of the is an interesting question that is outside the scope of this study. dispersion relation. The direction is determined by the phase of the ground state. We found two branches of gapless col- ACKNOWLEDGMENTS lective modes in this case that can be identified with the Nambu-Goldstone modes. Depending on the values of pa- We appreciate useful discussions with G. Aeppli, J. Bai- rameters, however, one or even both modes could become ley, A. Pertsova, A. Sinner, C. Triola, A. Yakimenko, and V. unstable for small values of wave vector. The case of a Zyuzin. This work was supported by the University of Con- maximally anisotropic distribution of magnons among the necticut, VILLUM FONDEN, via the Centre of Excellence sublattices is qualitatively different. In general, the spectrum for Dirac Materials (Grant No. 11744), the European Research of one of the branches is gapped and can be identified with the Council under the European Unions Seventh Framework Pro- Higgs or amplitude mode. For the certain range of parameters, gram Synergy HERO, and the Knut and Alice Wallenberg the modes in this case could be also unstable. It is worth noting Foundation KAW 2018.0104. P.S. acknowledges the support that while the case of a maximally anisotropic distribution of through the Yale Prize Postdoctoral Fellowship in Condensed magnons seems to be unlikely, it could have lower energy. Matter Theory. S.B. acknowledges support from Deutsche Independent of the ground state, the dynamics of collective Forschungsgemeinschaft (DFG, German Research Founda- modes could be effectively controlled via the Haldane gap tion) Grant No. TRR 80. term. This term might be intrinsically present in a system or generated and tuned via the effective pseudo-Zeeman effect in APPENDIX A: SCHEMATIC DERIVATION OF THE ferrimagnets. ENERGY DENSITY FOR DIRAC MAGNONS With the results presented in this paper, we hope that the search for the condensates in bosonic Dirac materials will In this Appendix, we provide a schematic derivation of the broaden. We reiterate the Dirac equation as a mathematical free-energy density (1) by using magnons on a honeycomb lat- structure that can be equally applied to fermions and bosons tice as a characteristic example. The Heisenberg Hamiltonian of the model reads as with specific symmetries [90]. As an explicit example, we  considered the case of Dirac magnons, whose experimental H =−J S · S , (A1) realization and condensation would be a new direction to pur- i j ij sue. Indeed, the majority of the studies in the field of magnon condensates are related to YIG that does not have Dirac points where the summation runs over the nearest-neighbor coupling in the magnon spectrum. Trihalides are the promising class of between spins Si. Further, J is the coupling constant, which is materials that can support Dirac magnons. A suitable class positive in the ferromagnetic phase.

013002-11 SUKHACHOV, BANERJEE, AND BALATSKY PHYSICAL REVIEW RESEARCH 3, 013002 (2021)

The position vectors of lattice sites for the sublattice A corresponding to the two sublattices A and B, respectively, we could be parametrized as follows: obtain   √ † = + , + x y aˆi aˆiaˆi −3/2 rn n1a1 n2a2 (A2) S ≡ S + iS = 2S aˆ − + O(S ), (A5) i i i i 4S   √ † † where the site label n = (n1, n2 ) is determined by a pair of aˆ aˆ aˆ S− ≡ Sx − iSy = 2S aˆ† − i i i + O(S−3/2 ), (A6) integers n1 and n2, and i i i i 4S √  √  z = − † . 3a 3a 3a 3a Si S aˆi aˆi (A7) a = , , a = , − (A3) 1 2 2 2 2 2 Similar expressions can be written for the sublattice B.By using these relations and performing the Fourier transform − / δ are the primitive translation vectors of the hexagonal lattice = 1 2 ik i aˆi N k e aˆk, where N is the total number of sites and a is the spacing between lattice sites. Further, the position and k is momentum, we derive the following Hamiltonian in vectors for the sublattice B are given by the leading-order approximation (see also Ref. [23]):   = δ + + , = ψˆ † ψˆ , rn 1 n1a1 n2a2 (A4) H k H0 k (A8) k δ = − / where 1 (a1 a2 ) 3 is the relative position of site B with where respect to site A in the unit cell. The relative positions of the   δ = δ + δ = 3 −γk other two sites of type B are given by 2 1 a2 and 3 H = JS ∗ , (A9) 0 −γ 3 δ1 − a1. k To bosonize Hamiltonian (A1), we use the standard   ψˆ = aˆk , Holstein-Primakoff transformation truncated to the first or- k ˆ (A10) der in the inverse spin 1/S. By introducing the bosonic bk ˆ † ˆ† annihilation (creation) operatorsa ˆi and bi (ˆai and bi ) and √     ik·δ iak − i ak 3akx γ = e j = e y + 2e 2 y cos . (A11) k 2 j The spectrum of Hamiltonian (A9) reads as √  √    3ak 3ak 3ak ± = JS(3 ±|γ |) = 3JS ± JS 1 + 4 cos x cos x + cos y . (A12) k k 2 2 2

As is easy to check, the two branches in Eq. (A12) touch at This magnon Hamiltonian has a structure of a massless Dirac ± = two nonequivalent√ isolated points in the Brillouin zone: kD Hamiltonian. Therefore, we call such quasiparticles Dirac ±4π/(3 3a)xˆ, where plus and minus signs correspond to K magnons. and K Dirac points or valleys, respectively. Here, rˆ = r/|r| Further, we add the term describing interaction with an  is the unit vector. In the vicinity of K and K points, i.e., at effective magnetic field Bi: k = k± + δk, where δk is small, the expression for ± takes  D k − μ · . the following simple form: g B (Bi Si ) (A17) ± i ζ  3JS ± v|δk|, (A13) k +δk D Note that we assume that Bi could be different at different where ζ =±is the valley degree of freedom and v = 3JSa/2 sublattices. Effectively, this might stem from slightly different is an analog of the Fermi velocity in graphene. Then, the gyromagnetic ratios g at the A and B sublattices, which indeed Hamiltonian in Eq. (A9) reads as occurs in ferrimagnets [27]. It is worth noting also that the pseudo-Zeeman effect was predicted for fermions in graphene (ζ ) = 1 + ζσ δ + σ δ . H0 3JS 2 v x kx v y ky (A14) in Ref. [80]. Its origin, however, is related to strains. For the sake of definiteness, we assume that Bi zˆ. Performing the Here, σ = (σx,σy ) is the vector of the Pauli matrices acting in the sublattice space. In terms of the bispinor Holstein-Primakoff transformation and omitting the constant term ∼gμBBS, we obtain the following term in the Hamilto- T ˆ δ = (ˆa+,δ , bˆ+,δ , bˆ−,δ , aˆ−,δ ) , (A15) nian: k k k k k  = μ ψˆ † 1 + ˜σ ψˆ . we have the following Hamiltonian in sublattice and valley HB g B k (B 2 B z ) k (A18) space:   k + δ · σ  ˆ † 3JS v( k )0ˆ . As one can see, the usual magnetic field simply shifts the spec- H0 δk δk 03JS − v(δk · σ) trum of magnons as 3JS → 3JS + gμBB. Therefore, it can (A16) be used to tune the effective chemical potential of magnons.

013002-12 BOSE-EINSTEIN CONDENSATE OF DIRAC MAGNONS: … PHYSICAL REVIEW RESEARCH 3, 013002 (2021)

TABLE I. Classification of Dirac condensates residing in two valleys of the spectrum shown in Fig. 1 based on their phase locking.

Density Type Phase Designation

θa,+ = θa,− θb,+ = θb,− Corotating na,+ = na,− Symmetric θa,+ =−θa,− θb,+ =−θb,− Anti corotating nb,+ = nb,− Corotating “B” condensate θa,+ =−θa,− θb,+ = θb,− Anti corotating “A” condensate na,+ = na,− Asymmetric Corotating “A” condensate θa,+ = θa,− θb,+ =−θb,− nb,+ = nb,− Anti corotating “B” condensate

Furthermore, it stabilizes the 2D Dirac ferromagnetic mate- rial. The effect of the sublattice-dependent field B˜ = (B − 3 2 a γ ζ +δ ≈− a(ζδkx − iδky ) + O(δk ). (A22) kD k Bb)/2 is qualitatively different. This term opens the gap in 2 the spectrum and is similar to the Zeeman effect albeit in the pseudospin space. Indeed, the energy spectrum (A12) reads as Further, we employ the Fourier transform of the operators

 ± = ± |γ |2 + μ ˜ 2/ 2 . k JS(3 k (g BB) (JS) ) (A19) 1 −ir·δk aˆζ,δk = √ dr e aˆζ,r, (A23) ˜ N The structure of the effective field B for Dirac magnons allows δ 1 d k ir·δk us to identify it with the Haldane gap term used in the main aˆζ, = √ e aˆζ,δ . (A24) r π 2 k text. The energy spectrum of Dirac magnons at B˜ = 0 and B˜ = N (2 ) 0 is shown in Figs. 1(a) and 1(b), respectively. The next-order Holstein-Primakoff transformation gives Note also that the following interaction term [23]:   (2π )2 J ∗ † † δ , = δ(k − k ). (A25) = δ + , + γ ˆ k1 k2 1 2 H4 k1 k2 k3 k4 k aˆk bk aˆk3 aˆk4 N 4N 2 1 2 k1,k2,k3,k4 + γ † † ˆ + γ ˆ† † ˆ ˆ + γ ∗ ˆ† ˆ† ˆ Then, the leading-order interaction term is k4 aˆk aˆk aˆk3 bk4 k2 bk aˆk bk3 bk4 k bk bk bk3 aˆk4 1 2 1 2 4 1 2 − 4γ ∗ aˆ† bˆ† aˆ bˆ . (A20)   k4−k2 k k k3 k4 3J † † 1 2 − δζ ,ζ ˆ ζ ,δ ζ ,δ H4 2 4 aˆζ ,δk bζ ,δk aˆ 3 k3 b 4 k4 Let us derive the interaction terms in the vicinity of the Dirac N 1 1 2 2 k1,k2,k3,k4 ζ1,ζ2,ζ3,ζ4  points and neglect the gradient terms. This approximation sig- 2 (2π ) − δ +δ −δ −δ · nificantly simplifies the final expression. Furthermore, these × dr e i( k1 k2 k4 k3 ) r N terms are expected to be weak for small deviations from the  Dirac nodes. We use  2 † † =−3J(2π ) dr aˆζ bˆζ aˆζ bˆζ . (A26) γ ≈ + δ 2 , 1 2 1 2 δk 3 O( k ) (A21) ζ1,ζ2

= . = . −1 = = = −1 FIG. 10. The dependence of the magnon densities on time t at a few values of uab for Pmax 0 5 P0, 0 1 t0 ,andga gb gab t0 . The densities are normalized to the total magnon density at t = tEnd. Here, the values of other parameters given in Eqs. (9)and(10)areused. = −2 / In addition, P0 is the characteristic strength of the pump (P0 t0 ). The Rabi oscillations, whose period is determined by 1 at small uab and > . −1 = . −1 large ( 0 1 t0 at uab 0 2 t0 ), are clearly visible for the magnon BECs.

013002-13 SUKHACHOV, BANERJEE, AND BALATSKY PHYSICAL REVIEW RESEARCH 3, 013002 (2021)

= . = . −1 = = FIG. 11. The dependence of the magnon densities on time t at a few values of uab for Pmax 0 5 P0, 0 1 t0 , ga gb 0, and =−−1 = gab t0 . The densities are normalized to the total magnon density at t tEnd. Here, the values of other parameters given in Eqs. (9) = −2 and (10) are used. In addition, P0 is the characteristic strength of the pump (P0 t0 ). The Rabi oscillations, whose period is determined by / > . −1 = . −1 1 at small uab and large ( 0 1 t0 at uab 0 2 t0 ), are clearly visible for the magnon BECs.

Therefore, by using the Bogolyubov approximation for the worth noting also that we considered a simple model for Dirac field operators [52], e.g.,a ˆ → ψa,in magnons in the derivation above. Other interaction terms,     such as terms corresponding to the Dzyaloshinskii-Moriya † † † and Kitaev interactions, were omitted in this schematic H = dr 3JS [ˆaζ aˆζ + bˆζ bˆζ ] − iv aˆζ [ζ∂x − i∂y]bˆζ ζ ζ derivation. However, their effects can be still included phe-   nomenologically, at least in part, by adding additional terms to † 2 † † − iv bˆ [ζ∂ + i∂ ]ˆaζ − 3J(2π ) aˆ bˆ aˆζ bˆζ the free energy (1). The corresponding microscopical analysis ζ x y ζ1 ζ2 1 2 ζ ζ ,ζ will be reported elsewhere.  1 2  † † + [ˆaζ aˆζ − bˆζ bˆζ ] , (A27) ζ APPENDIX B: CLASSIFICATION OF NONTRIVIAL PHASES OF DIRAC BEC we obtain the final expression for the energy density   In this Appendix, we discuss possible nontrivial phases of = − ψ∗ ζ∂ − ∂ ψ − ψ∗ ζ∂ c0ntot iv a,ζ [ x i y] b,ζ iv b,ζ [ x the Dirac BECs formed at the two valleys of the Brillouin ζ ζ zone. In the absence of interaction, the symmetry class for the BEC corresponds to U(1) + × U(1) − × U(1) + × U(1) −, + i∂y]ψa,ζ + gabnanb + (na − nb). (A28) a a b b  where each of the symmetries is related to the independent = ψ∗ ψ = ψ∗ ψ = Here,na,ζ a,ζ a,ζ , nb,ζ b,ζ b,ζ , na ζ =± na,ζ , phase rotation of the wave function. We denote the wave = = + ψ α = , ,ζ =± nb ζ =± nb,ζ , and ntot na nb. This energy density functions α,ζ ( a b √) at each valley with the pseu- iθα,ζ serves as a starting point for the free energy in Eq. (1). It is dospin components as ψα,ζ = nα,ζ e . Interaction could

= . −1 = = = −1 FIG. 12. The dependence of the magnon densities on time t at a few values of Pmax for 0 1 t0 and ga gb gab t0 . The densities = = −2 are normalized to the total magnon density at t tEnd. Here, P0 is the characteristic strength of the pump (P0 t0 ).

013002-14 BOSE-EINSTEIN CONDENSATE OF DIRAC MAGNONS: … PHYSICAL REVIEW RESEARCH 3, 013002 (2021)

FIG. 13. Frequencies of the collective modes ω+ (a) and ω− (b) for nb = 0 at a few values of the Haldane gap . Here, k0 = 1/(vt0 )and ω0 = vk0. Shaded regions denote the imaginary part of the frequencies.

lead to phase-locked condensates, a few of which are provided different phase locking can appear as θa,+ =−θa,− and θb,+ = in Table I. θb,− (or θa,+ = θa,− and θb,+ =−θb,−). In this case, we define For simplicity we focus on the possible phase lockings it as anticorotating “A” (“B”) and corotating “B” (“A”). between the condensates at two nonequivalent Dirac points. The two amplitudes na and nb at the two different valleys can be equal to each other or might be different. In the former APPENDIX C: EFFECT OF INTERCONDENSATE case, we designate the condensates as symmetric and in the INTERACTIONS ON THE CONDENSATE DYNAMICS latter case, they are asymmetric. For each of these ampli- Let us consider the effect of nonzero interaction constant tude classes, we then classify the condensates based on the g on the condensate dynamics. By using the model defined phase relations. If the phases of condensate components are ab in Sec. III A in the main text, we numerically calculate the equal in magnitude at the two valleys as θ / ,+ = θ / ,−,we − a b a b magnon densities n , n , and n for g = g = g = t 1 (see call them corotating. In the other case, when the phases are a b 3 a b ab 0 g = g = g =−t −1 equal in magnitude but differ in sign, the condensates are Fig. 10)aswellas a b 0 and ab 0 (see Fig. 11). called anticorotating. However, for suitable interaction chan- Note that the interaction constants ga, gb, gab are not man- nels between the multicomponent condensates, a completely ifested in the condensate dynamics when the phase-mixing terms are absent (e.g., the Josephson coupling term uab). As

FIG. 14. Frequencies of the collective modes ω+ (a) and ω− (b) for nb = 0 at a few values of the Haldane gap . Here, k0 = 1/(vt0 ), ω0 = vk0,andcr =|(gb − gab)na|/2. Shaded regions denote the imaginary part of the frequencies. Except a single point = cr, one of the modes is gapped and the other is gapless.

013002-15 SUKHACHOV, BANERJEE, AND BALATSKY PHYSICAL REVIEW RESEARCH 3, 013002 (2021) one can see by comparing Fig. 4 with Figs. 10 and 11,the On the other hand, the lowest branch can acquire magnons via effect of the intercondensate interaction constant gab is similar thermalization of magnons at the Dirac points. For example, to that of ga and gb at small uab. On the other hand, the this could be achieved via four-magnon processes (in the case anharmonic oscillations of the condensates are more easily of YIG, see, e.g., Ref. [68]). This process is described via the accessible for a nonzero gab. terms with P0b and P0a. By using the values of parameters in Sec. III B as well as assuming that = = = = 0.25 t −1 and P = APPENDIX D: FOUR-COMPONENT MODEL 0 a b 3 0 0a P0b = 0.5, we present the results for the population densities To describe the magnon BECs at the Dirac nodes and at the in Fig. 12. As one can see, the magnon density at the point point, we extend the phenomenological model in Sec. III A. raises at the later stages of the pumping. Since the source of In addition to Eqs. (5)–(7), a Gross-Pitaevskii equation for the these magnons is the magnon BEC at the Dirac points, the magnon population at k = 0 is included. The full system then rise of the magnon density at the lowest branch leads to the reads as depletion of the Dirac magnons. Another effect that would be interesting to test experimentally and/or verify microscopi- i∂t ψa + iaψa = (c0 − μ + )ψa − iv[(∂x − i∂y )ψb] cally is the rise of the pumped magnon density. In the model at ∗ hand, this fact stems from the decrease of the Dirac magnons’ + g n ψ + g n ψ + u ψ (ψ )2 a a a ab b a ab a b densities. + iPa3an3ψa − iPa00n0ψa, (D1) APPENDIX E: COLLECTIVE MODES WITH JOSEPHSON ∂ ψ + ψ = − μ − ψ − ∂ + ∂ ψ i t b i b b (c0 ) b iv[( x i y ) a] COUPLING TERM + ψ + ψ + ∗ ψ∗ ψ 2 gbnb b gabna b uab b ( a ) Let us briefly discuss the effect of the Josephson coupling

+ iPb3bn3ψb − iPb00n0ψb, (D2) term uab on collective modes considered in Sec. IV.The coupling terms complicate the dynamics of the system and 1 ∂ ψ + ψ =− ∂2 + ∂2 ψ + ψ do not allow for a simple analytical solution. Therefore, the i t 0 i 0 0 x y 0 g0n0 0 2meff frequencies of the collective modes are obtained by solv- + iP n ψ + iP n ψ , (D3) ing the system of equations (23)to(26) numerically with 0b a a 0 0a b b 0 = = = −1 =  c0 ga gb t0 , gab 0, and the ground state defined in Sec. IV A. The corresponding results for u = 0.2 t −1 are ∂t n3 + 3n3 = 3P˜(t ) − (P3aana,ζ + P3bbnb,ζ )n3. ab 0 ζ =± shown in Figs. 13 and 14 at nb = 0 and nb = 0, respectively. (D4) As one can see, the imaginary part is enhanced compared to the case uab considered in Sec. IV C. Moreover, since we Here, the magnon BEC at the point is described by Eq. (D3). no longer have independent phases for the ground-state wave Due to kinematic constraints, it is unlikely that magnons can functions at nb = 0, one of the modes can become gapped [see scatter to the point directly from the pumped population. Fig. 13(a)].

[1] M. I. Katsnelson, Graphene: in two dimensions, Mater. [11] B. Yan and C. Felser, Topological materials: Weyl , Today 10, 20 (2007). Annu. Rev. Condens. Matter Phys. 8, 337 (2017). [2] A. K. Geim and K. S. Novoselov, The rise of graphene, Nat. [12] M. Z. Hasan, S.-Y. Xu, I. Belopolski, and C.-M. Huang, Dis- Mater. 6, 183 (2007). covery of Weyl fermion semimetals and topological Fermi arc [3] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, states, Annu. Rev. Condens. Mattter Phys. 8, 289 (2017). and A. K. Geim, The electronic properties of graphene, Rev. [13] N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and Dirac Mod. Phys. 81, 109 (2009). semimetals in three-dimensional solids, Rev. Mod. Phys. 90, [4] A. K. Geim, Graphene: Status and prospects, Science 324, 1530 015001 (2018). (2009). [14] T. Jacqmin, I. Carusotto, I. Sagnes, M. Abbarchi, D. D. [5] M. I. Katsnelson, Graphene: Carbon in Two Dimensions (Cam- Solnyshkov, G. Malpuech, E. Galopin, A. Lemaître, J. Bloch, bridge University Press, Cambridge, 2012). and A. Amo, Direct Observation of Dirac Cones and a Flatband [6] M. Z. Hasan and C. L. Kane, Colloquium: Topological insula- in a Honeycomb Lattice for Polaritons, Phys. Rev. Lett. 112, tors, Rev. Mod. Phys. 82, 3045 (2010). 116402 (2014). [7] X.-L. Qi and S.-C. Zhang, Topological insulators and supercon- [15] L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, ductors, Rev. Mod. Phys. 83, 1057 (2011). M. Soljaciˇ c,´ Experimental observation of Weyl points, Science [8] B. A. Bernevig and T. L. Hughes, Topological Insulators 349, 622 (2015). and Topological Superconductors (Princeton University Press, [16] H.-X. Wang, L. Xu, H.-Y. Chen, and J.-H. Jiang, Three- Princeton, NJ, 2013). dimensional photonic Dirac points stabilized by point group [9] A. M. Turner and A. Vishwanath, Beyond Band Insulators: symmetry, Phys.Rev.B93, 235155 (2016). Topology of Semimetals and Interacting Phases (Elsevier, Am- [17] B. Yang, Q. Guo, B. Tremain, R. Liu, L. E. Barr, Q. Yan, W. sterdam, 2013). Gao, H. Liu, Y. Xiang, J. Chen, C. Fang, A. Hibbins, L. Lu, [10] T. O. Wehling, A. M. Black-Schaffer, and A. V. Balatsky, Dirac and S. Zhang, Ideal Weyl points and helicoid surface states in materials, Adv. Phys. 63, 1 (2014). artificial photonic crystal structures, Science 359, 1013 (2018).

013002-16 BOSE-EINSTEIN CONDENSATE OF DIRAC MAGNONS: … PHYSICAL REVIEW RESEARCH 3, 013002 (2021)

[18] Q. Guo, O. You, B. Yang, J. B. Sellman, E. Blythe, H. Liu, [35] W. Yao, C. Li, L. Wang, S. Xue, Y. Dan, K. Iida, K. Kamazawa, Y. Xiang, J. Li, D. Fan, J. Chen, C. T. Chan, and S. Zhang, K. Li, C. Fang, and Y.Li, Topological spin excitations in a three- Observation of Three-Dimensional Photonic Dirac Points and dimensional antiferromagnet, Nat. Phys. 14, 1011 (2018). Spin-Polarized Surface Arcs, Phys. Rev. Lett. 122, 203903 [36] L.-C. Zhang, Y. A. Onykiienko, P. M. Buhl, Y. V. Tymoshenko, (2019). P. Cermák,ˇ A. Schneidewind, J. R. Stewart, A. Henschel, M. [19] Y. Yang, Z. Gao, H. Xue, L. Zhang, M. He, Z. Yang, R. Singh, Schmidt, S. Blügel, D. S. Inosov, and Y. Mokrousov, Magnonic

Y. Chong, B. Zhang, and H. Chen, Realization of a three- Weyl states in Cu2OSeO3, Phys. Rev. Research 2, 013063 dimensional photonic , Nature (London) (2020). 565, 622 (2019). [37] S. Banerjee, D. Abergel, H. Agren, G. Aeppli, and A. V. [20] M. Xiao, W.-J. Chen, W.-Y. He, and C. T. Chan, Synthetic gauge Balatsky, Interacting Dirac materials, J. Phys.: Condens. Matter flux and Weyl points in acoustic systems, Nat. Phys. 11, 920 32, 405603 (2020). (2015). [38] L. Zhang, J. Ren, J.-S. Wang, and B. Li, Topological magnon [21] M. Serra-Garcia, V. Peri, R. Süsstrunk, O. R. Bilal, T. Larsen, insulator in insulating ferromagnet, Phys. Rev. B 87, 144101 L. G. Villanueva, and S. D. Huber, Observation of a phononic (2013). quadrupole topological insulator, Nature (London) 555, 342 [39] A. Mook, J. Henk, and I. Mertig, Edge states in topological (2018). magnon insulators, Phys.Rev.B90, 024412 (2014). [22] X. Cai, L. Ye, C. Qiu, M. Xiao, R. Yu, M. Ke, and Z. Liu, [40] S. A. Owerre, A first theoretical realization of honeycomb topo- Symmetry-enforced three-dimensional Dirac phononic crystals, logical magnon insulator, J. Phys.: Condens. Matter 28, 386001 Light Sci. Appl. 9, 38 (2020). (2016). [23] S. S. Pershoguba, S. Banerjee, J. C. Lashley, J. Park, H. Ågren, [41] K. Nakata, J. Klinovaja, and D. Loss, Magnonic quantum Hall G. Aeppli, and A. V. Balatsky, Dirac Magnons in Honeycomb effect and Wiedemann-Franz law, Phys. Rev. B 95, 125429 Ferromagnets, Phys.Rev.X8, 011010 (2018). (2017). [24] L. Chen, J.-H. Chung, B. Gao, T. Chen, M. B. Stone, A. I. [42] A. A. Kovalev, V. A. Zyuzin, and B. Li, Pumping of magnons in Kolesnikov, Q. Huang, and P. Dai, Topological Spin Excita- a Dzyaloshinskii-Moriya ferromagnet, Phys.Rev.B95, 165106

tions in Honeycomb Ferromagnet CrI3, Phys. Rev. X 8, 041028 (2017). (2018). [43] S. A. Owerre, Magnonic analogs of topological Dirac semimet- [25] I. Lee, F. G. Utermohlen, D. Weber, K. Hwang, C. Zhang, J. van als, J. Phys. Commun. 1, 025007 (2017). Tol, J. E. Goldberger, N. Trivedi, and P. C. Hammel, Funda- [44] X. S. Wang, A. Brataas, and R. E. Troncoso, Bosonic Bott In- mental Spin Interactions Underlying the Magnetic Anisotropy dex and Disorder-Induced Topological Transitions of Magnons,

in the Kitaev Ferromagnet CrI3, Phys.Rev.Lett.124, 017201 Phys. Rev. Lett. 125, 217202 (2020). (2020). [45] Y. Ferreiros and M. A. H. Vozmediano, Elastic gauge fields [26] S. Banerjee, J. Fransson, A. M. Black-Schaffer, H. Ågren, and and Hall viscosity of Dirac magnons, Phys. Rev. B 97, 054404 A. V. Balatsky, Granular superconductor in a honeycomb lattice (2018). as a realization of bosonic Dirac material, Phys. Rev. B 93, [46] M. M. Nayga, S. Rachel, and M. Vojta, Magnon Landau Levels 134502 (2016). and Emergent Supersymmetry in Strained Antiferromagnets, [27] J. Fransson, A. M. Black-Schaffer, and A. V. Balatsky, Magnon Phys. Rev. Lett. 123, 207204 (2019). Dirac materials, Phys.Rev.B94, 075401 (2016). [47] T. Liu and Z. Shi, Twist-induced magnon Landau levels in [28] C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao, W. Bao, honeycomb magnets, arXiv:2004.01371. C. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava, S. G. Louie, J. Xia, [48] P. A. McClarty and J. G. Rau, Non-Hermitian topology of spon- and X. Zhang, Discovery of intrinsic ferromagnetism in two- taneous magnon decay, Phys.Rev.B100, 100405(R) (2019). dimensional van der Waals crystals, Nature (London) 546, 265 [49] A. Pertsova and A. V. Balatsky, Dynamically induced excitonic (2017). instability in pumped Dirac materials, Ann. Phys. (Berlin) 532, [29] B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. 1900549 (2020). Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. McGuire, [50] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, The- D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-Herrero, and X. Xu, ory of Bose-Einstein condensation in trapped gases, Rev. Mod. Layer-dependent ferromagnetism in a van der Waals crystal Phys. 71, 463 (1999). down to the monolayer limit, Nature (London) 546, 270 (2017). [51] R. Ozeri, N. Katz, J. Steinhauer, and N. Davidson, Colloquium: [30] A. C. Gossard, V. Jaccarino, and J. P. Remeika, Experimental Bulk Bogoliubov excitations in a Bose-Einstein condensate, Test of the Spin-Wave Theory of a Ferromagnet, Phys. Rev. Rev. Mod. Phys. 77, 187 (2005). Lett. 7, 122 (1961). [52] L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation [31] H. L. Davis and A. Narath, Spin-wave renormalization applied and Superfluidity (Oxford University Press, Oxford, 2016).

to ferromagnetic CrBr3, Phys. Rev. 134, A433 (1964). [53] L. H. Haddad and L. D. Carr, The nonlinear Dirac equa- [32] F.-Y. Li, Y.-D. Li, Y. B. Kim, L. Balents, Y. Yu, and G. Chen, tion in Bose–Einstein condensates: Foundation and symmetries, Weyl magnons in breathing pyrochlore antiferromagnets, Nat. Physica D (Amsterdam) 238, 1413 (2009). Commun. 7, 12691 (2016). [54] L. H. Haddad and L. D. Carr, Relativistic linear stability [33] A. Mook, J. Henk, and I. Mertig, Tunable Magnon Weyl Points equations for the nonlinear Dirac equation in Bose-Einstein in Ferromagnetic Pyrochlores, Phys. Rev. Lett. 117, 157204 condensates, Europhys. Lett. 94, 56002 (2011). (2016). [55] L. H. Haddad, K. M. O’Hara, and L. D. Carr, Nonlinear Dirac [34] S. A. Owerre, Weyl magnons in noncoplanar stacked kagome equation in Bose-Einstein condensates: Preparation and stabil- antiferromagnets, Phys. Rev. B 97, 094412 (2018). ity of relativistic vortices, Phys.Rev.A91, 043609 (2015).

013002-17 SUKHACHOV, BANERJEE, AND BALATSKY PHYSICAL REVIEW RESEARCH 3, 013002 (2021)

[56] L. H. Haddad, C. M. Weaver, and L. D. Carr, The nonlinear Controlling of nonlinear relaxation of quantized magnons in Dirac equation in Bose–Einstein condensates: I. Relativistic nano-devices, arXiv:2006.03400. solitons in armchair nanoribbon optical lattices, New J. Phys. [72] J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, 17, 063033 (2015). J. M. J. Keeling, F. M. Marchetti, M. H. Szymanska,´ R. André, [57] L. H. Haddad and L. D. Carr, The nonlinear Dirac equation J. L. Staehli, V. Savona, P. B. Littlewood, B. Deveaud, and in Bose–Einstein condensates: II. Relativistic soliton stability L. S. Dang, Bose–Einstein condensation of exciton polaritons, analysis, New J. Phys. 17, 063034 (2015). Nature (London) 443, 409 (2006). [58] L. H. Haddad and L. D. Carr, The nonlinear Dirac equation in [73] H. Deng, H. Haug, and Y. Yamamoto, Exciton- Bose- Bose–Einstein condensates: Vortex solutions and spectra in a Einstein condensation, Rev. Mod. Phys. 82, 1489 (2010). weak harmonic trap, New J. Phys. 17, 113011 (2015). [74] B. Deveaud, Exciton-polariton Bose-Einstein condensates, [59] L. H. Haddad and L. D. Carr, The nonlinear Dirac equation in Annu. Rev. Condens. Matter Phys. 6, 155 (2015) Bose–Einstein condensates: Superfluid fluctuations and emer- [75] Y. Sun, P. Wen, Y. Yoon, G. Liu, M. Steger, L. N. Pfeiffer, K. gent theories from relativistic linear stability equations, New J. West, D. W. Snoke, and K. A. Nelson, Bose-Einstein Condensa- Phys. 17, 093037 (2015). tion of Long-Lifetime Polaritons in Thermal Equilibrium, Phys. [60] J. Cuevas-Maraver, P. G. Kevrekidis, A. Saxena, A. Comech, Rev. Lett. 118, 016602 (2017). and R. Lan, Stability of Solitary Waves and Vortices in a 2D [76] Universal Themes of Bose-Einstein Condensation, edited by Nonlinear Dirac Model, Phys.Rev.Lett.116, 214101 (2016). N. P. Proukakis, D. W. Snoke, and P. B. Littlewood (Cambridge [61] Nonlinear Systems, Vol. 1. Understanding Complex Systems, University Press, Cambridge, 2017). edited by V. Carmona, J. Cuevas-Maraver, F. Fernández- [77] V. S. L’vov, Wave Turbulence under Parametric Excitation: Sánchez, and E. García- Medina (Springer, Cham, 2018). Applications to Magnets (Springer, Berlin, 1994). [62] A. N. Poddubny and D. A. Smirnova, Ring Dirac solitons in [78] E. V. Gorbar, V. A. Miransky, and I. A. Shovkovy, Chiral nonlinear topological systems, Phys. Rev. A 98, 013827 (2018). asymmetry of the Fermi surface in dense relativistic matter in [63] S. O. Demokritov, V. E. Demidov, O. Dzyapko, G. A. Melkov, a magnetic field, Phys. Rev. C 80, 032801(R) (2009). A. A. Serga, B. Hillebrands, and A. N. Slavin, Bose–Einstein [79] Y. J. Lin, K. Jimenez-Garcia, and I. B. Spielman, Spin–orbit- condensation of quasi-equilibrium magnons at room tempera- coupled Bose-Einstein condensates, Nature (London) 471,83 ture under pumping, Nature (London) 443, 430 (2006). (2011). [64] V. E. Demidov, O. Dzyapko, S. O. Demokritov, G. A. Melkov, [80] J. L. Mañes, F. de Juan, M. Sturla, and M. A. H. Vozmediano, and A. N. Slavin, Thermalization of a Parametrically Driven Generalized effective Hamiltonian for graphene under nonuni- Magnon Gas Leading to Bose-Einstein Condensation, Phys. form strain, Phys.Rev.B88, 155405 (2013). Rev. Lett. 99, 037205 (2007). [81] V. Hahn and P. Kopietz, Collisionless kinetic theory for para- [65] O. Dzyapko, V. E. Demidov, S. O. Demokritov, G. A. Melkov, metrically pumped magnons, Eur. Phys. J. B 93, 132 (2020). and A. N. Slavin, Direct observation of Bose-Einstein conden- [82] S. O. Demokritov, V. E. Demidov, O. Dzyapko, G. A. Melkov, sation in a parametrically driven gas of magnons, New J. Phys. and A. N. Slavin, Quantum coherence due to Bose-Einstein 9, 64 (2007). condensation of parametrically driven magnons, New J. Phys. [66] V. E. Demidov, O. Dzyapko, S. O. Demokritov, G. A. Melkov, 10, 045029 (2008). and A. N. Slavin, Observation of Spontaneous Coherence in [83] S. M. Rezende, Theory of coherence in Bose-Einstein conden- Bose-Einstein Condensate of Magnons, Phys.Rev.Lett.100, sation phenomena in a microwave-driven interacting magnon 047205 (2008). gas, Phys. Rev. B 79, 174411 (2009). [67] P. Nowik-Boltyk, O. Dzyapko, V. E. Demidov, N. G. Berloff, [84] S. M. Rezende, Wave function of a microwave-driven Bose- and S. O. Demokritov, Spatially non-uniform ground state and Einstein magnon condensate, Phys. Rev. B 81, 020414(R) quantized vortices in a two-component Bose-Einstein conden- (2010). sate of magnons, Sci. Rep. 2, 482 (2012). [85] D. Prananto, Y. Kainuma, K. Hayashi, N. Mizuochi, K. Uchida, [68] A. A. Serga, V. S. Tiberkevich, C. W. Sandweg, V. I. Vasyuchka, and T. An, Probing thermal magnon current via nitrogen- D. A. Bozhko, A. V. Chumak, T. Neumann, B. Obry, G. A. vacancy centers in diamond, arXiv:2007.13433. Melkov, A. N. Slavin, and B. Hillebrands, Bose–Einstein [86] V. V. Kruglyak, S. O. Demokritov, and D. Grundler, Magnonics, condensation in an ultra-hot gas of pumped magnons, Nat. J. Phys. D: Appl. Phys. 43, 264001 (2010). Commun. 5, 3452 (2014). [87] See Supplemental Material at http://link.aps.org/supplemental [69] D. A. Bozhko, A. J. E. Kreil, H. Yu. Musiienko-Shmarova, /10.1103/PhysRevResearch.3.013002 for the animation of dy- A. A. Serga, A. Pomyalov, V. S. L’vov, and B. Hillebrands, namics of the Dirac magnon populations. Bogoliubov waves and distant transport of magnon condensate [88] E. Fradkin, S. A. Kivelson, M. J. Lawler, J. P. Eisenstein, and at room temperature, Nat. Commun. 10, 2460 (2019). A. P. Mackenzie, Nematic fermi fluids in condensed matter [70] M. Mohseni, A. Qaiumzadeh, A. A. Serga, A. Brataas, physics, Annu. Rev. Condens. Matter Phys. 1, 153 (2010). B. Hillebrands, and P. Pirro, Bose-Einstein condensation of [89] C.-M. Jian and H. Zhai, Paired superfluidity and fractionalized nonequilibrium magnons in laterally confined systems, New J. vortices in systems of spin-orbit coupled bosons, Phys.Rev.B Phys. 22, 083080 (2020) 84, 060508(R) (2011). [71] M. Mohseni, Q. Wang, B. Heinz, M. Kewenig, M. Schneider, [90] M. Larsson and A. Balatsky, Paul Dirac and the Nobel Prize in F. Kohl, B. Lägel, C. Dubs, A. V. Chumak, and P. Pirro, physics, Phys. Today 72(11), 46 (2019).

013002-18