PHYSICAL REVIEW RESEARCH 3, 013002 (2021)
Bose-Einstein condensate of Dirac magnons: 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 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.
DOI: 10.1103/PhysRevResearch.3.013002
I. INTRODUCTION two-dimensional (2D) honeycomb lattice [23,27]. The cor- responding magnon 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 polaritons 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 Fermi surface exists in bosonic Dirac materials. For the excitations like magnons or polaritons, the equi- librium chemical potential 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 fermions, a nonvanishing population cannot be investigating various Dirac BECs. Since no Fermi surface achieved via equilibrium doping 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 Dirac equation, 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 quasiparticles. 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 wave function 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 Pauli matrices in four degrees of freedom: two pseudospins and two valley or τ the pseudospin space, z is the Pauli matrix 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 graphene, 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 + i aψa = (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 + iPa3 an3ψa, (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 + i bψb = (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 + iPb3 bn3ψb, (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 ) − (P3a ana,ζ + P3b bnb,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-phonon 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.
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= . = . −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 fermion 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 <