Parametric pumping and kinetics of magnons in dipolar ferromagnets
PParametric pumping and kinetics of magnons in dipolar ferromagnets
Thomas Kloss, Andreas Kreisel, and Peter Kopietz
Institut f¨ur Theoretische Physik, Universit¨at Frankfurt,Max-von-Laue Strasse 1, 60438 Frankfurt, Germany (Dated: April 8, 2010)The time evolution of magnons subject to a time-dependent microwave field is usually describedwithin the so-called “S-theory”, where kinetic equations for the distribution function are obtainedwithin the time-dependent Hartree-Fock approximation. To explain the recent observation of“Bose-Einstein condensation of magnons” in an external microwave field [Demokritov et al. , Na-ture , 430 (2006)], we extend the “S-theory” to include the Gross-Pitaevskii equation for thetime-dependent expectation values of the magnon creation and annihilation operators. We explicitlysolve the resulting coupled equations within a simple approximation where only a single condensedmode is retained. We also re-examine the usual derivation of an effective boson model from a realis-tic spin model for yttrium-iron garnet films and argue that in the parallel pumping geometry (whereboth the static and the time-dependent magnetic field are parallel to the macroscopic magnetization)the time-dependent Zeemann energy cannot give rise to magnon condensation.
PACS numbers: 75.30.Ds, 76.20.+q, 03.75.Kk
I. INTRODUCTION
When ordered magnets are exposed to microwave radi-ation of sufficiently high power, one typically observes anexponential growth of the population of certain groupsof spin-wave modes during some intermediate time in-terval. This is an example for a general phenomenonwhich is usually referred to as parametric resonance .A particularly suitable system for observing parametricresonance are yttrium-iron garnet (YIG) crystals, be-cause the spin-waves in this system have a very lowdamping. Early microscopic theories explaining para-metric resonance in magnetic insulators have been de-veloped by Suhl, and by Schl¨omann and co-authors. In the 1970s Zakharov, L’vov, and Starobinets devel-oped a comprehensive kinetic theory of parametric res-onance in magnon gases which is sometimes called “S-theory”. In this approach kinetic equations for the time-dependent distribution functions n k ( t ) = (cid:104) a † k ( t ) a k ( t ) (cid:105) and p k ( t ) = (cid:104) a − k ( t ) a k ( t ) (cid:105) are derived within the self-consistent time-dependent Hartree-Fock approximation.Here a k ( t ) and a † k ( t ) are the annihilation and creation op-erators of magnons with momentum k in the Heisenbergpicture. Subsequently the non-linear kinetic equations ofthe “S-theory” and extensions thereof have been studiedby many authors. Quite recently Demokritov and co-workers ob-served a new coherence effect of magnons in YIG un-der the influence of an external microwave field whichthey interpreted as Bose-Einstein condensation (BEC)of magnons at room temperature. A similar phe-nomenon has been observed in superfluid He, whereNMR pumping can cause the magnetization to precessphase-coherently. The emergence of this coherent statecan also be viewed as magnon BEC.
Whether ornot the experiments by Demokritov et al. can beconsidered to be an analogue of BEC in atomic Bosegases (which nowadays is routinely realized using ultra- cold atoms in an optical trap) has been discussed con-troversially in the literature.
We argue below thatthe coherent state generated in these experiments should perhaps not be called a Bose-Einstein conden-sate, because the condensation is not accompanied byspontaneous symmetry breaking in this case; instead,the microwave field gives rise to a term in the hamil-tonian which explicitly breaks the U (1)-symmetry of themagnon hamiltonian.Unfortunately, the conventional “S-theory” is insuffi-cient to describe the experimental situation, because thecoherent magnon state generated in the experiments ischaracterized by finite expectation values of the magnonannihilation and creation operators a k ( t ) and a † k ( t ) forcertain special values of k . In the condensed phase, thekinetic equations for the pair correlators n k ( t ) and p k ( t )should therefore be augmented by equations of motionfor the expectation values (cid:104) a k ( t ) (cid:105) and (cid:104) a † k ( t ) (cid:105) . Recallthat in the theory of the interacting Bose gas the corre-sponding equation of motion for the order-parameter iscalled Gross-Pitaevskii equation; this equation is miss-ing in the conventional “S-theory” which therefore doesnot completely describe the coherent magnon state inthe regime of strong pumping. In this work we shall out-line an extension of “S-theory” which includes the orderparameter dynamics on equal footing with the kineticequations for the distribution functions. Since we wouldlike to clarify conceptual points rather than performingexplicit quantitative calculations, we shall derive our ex-tended “S-theory” within the framework of a simple toymodel which we motivate in the following section. II. TOY MODEL FOR PARAMETRICRESONANCE IN YIG
In order to understand a complex physical phe-nomenon, it is sometimes useful to study a simplified a r X i v : . [ c ond - m a t . s t r- e l ] A p r “toy model” which still contains some essential featuresof the phenomenon of interest. For our purpose, it is suf-ficient to consider a single anharmonic oscillator with anadditional time-dependent term describing the creationand annihilation of pairs of particles. The hamiltonian isˆ H ( t ) = (cid:15) a † a + γ e − iω t a † a † + γ ∗ e iω t aa + u a † a † aa. (1)Here a and a † are bosonic annihilation and creation op-erators, (cid:15) > u > ω > γ to themagnon gas. Below we shall show that this model con-tains the essential physics of parametric resonance andBEC of magnons; in particular, in the regime of strongpumping | γ | > | (cid:15) − ω / | the model has a stationarynon-equilibrium state which corresponds to the coherentmagnon state observed in the experiments by Demokri-tov and co-workers. Our toy model (1) involves only a single boson operatorrepresenting the magnon at the minimum of the disper-sion which is expected to condense. Of course, for exper-imentally relevant macroscopic samples of YIG a morerealistic model should describe infinitely many magnonoperators a k labeled by crystal-momentum k , so that thefollowing bosonic “resonance hamiltonian” should give abetter description of the experimental situation,ˆ H res ( t ) = (cid:88) k (cid:15) k a † k a k + 12 (cid:88) k (cid:104) γ k e − iω t a † k a †− k + γ ∗ k e iω t a − k a k (cid:105) + 12 (cid:88) k , k (cid:48) , q u ( k , k (cid:48) , q ) a † k + q a † k (cid:48) − q a k (cid:48) a k . (2)If we assume that the k = 0 boson condenses and re-tain only this degree of freedom on the right-hand sideof Eq. (2), we arrive at our toy model (1). In the the-ory of superfluidity a similar reduced description involv-ing only the order parameter is provided by the Gross-Pitaevskii equation. Of course, the minimum of the dis-persion in experimentally relevant samples of YIG occursat certain non-zero wave-vectors ± k ∗ , so that it would bemore accurate to retain the two modes a k ∗ and a − k ∗ andtheir mutual interactions in Eq. (2). Moreover, the factthat in the experiments the wave-vectors of the con-densed magnons are different from the wave-vectors ofthe magnons which are initially generated by microwavepumping cannot be described within the framework ofour toy model. Nevertheless, below we shall show thatour simple model allows us to understand some concep-tual points related to the nature of the coherent stateobserved in the experiments. The bosonic resonance hamiltonian (2) has been thestarting point of several theoretical investigations ofparametric resonance in magnon gases.
This modelis believed to be a realistic model for YIG in the par-allel pumping geometry, where the static and the time-dependent components of the external magnetic fields areboth parallel to the direction of the macroscopic magneti-zation. In the appendix we shall critically re-examine theusual derivation of Eq. (2) from an effective spin hamil-tonian for YIG and show that in spin language the time-dependent resonance term in the second line of Eq. (2)involves also the combinations cos( ω t )[ S xi S xi − S yi S yi ] andsin( ω t )[ S xi S yi + S yi S xi ], where S αi are the components ofthe spin operators at lattice site i . Terms of this typecannot be related to the Zeemann energy associated witha time-dependent magnetic field parallel to the magne-tization. This is obvious for a ferromagnet with onlyexchange interactions, because in this case the magnonoperators a k and a † k can be identified with the Fouriercomponents of the Holstein-Primakoff bosons a i and a † i , which in turn can be related to the usual spin ladderoperators S + i and S − i ; to leading order for large spin S , S + i ≈ √ Sa i , S − i ≈ √ Sa † i . (3)Note, however, that the spin Hilbert space has only 2 S +1states per site, whereas the bosonic Fock space associatedwith the canonical boson operators a i and a † i is infinitedimensional; the identification of magnons with canoni-cal bosons is therefore only approximate. For a descrip-tion of coherence phenomena involving large occupan-cies of magnon states one should therefore keep in mindthat there is a constraint on the magnon Hilbert space.Assuming for simplicity that the parameter γ k = γ inEq. (2) is real and independent of k , the second term onthe right-hand side of Eq. (2) can be written as γ (cid:88) k (cid:104) e − iω t a † k a †− k + e iω t a − k a k (cid:105) ≈ γ S (cid:88) i (cid:2) e − iω t S − i S − i + e iω t S + i S + i (cid:3) = γ S (cid:88) i (cid:110) cos( ω t ) [ S xi S xi − S yi S yi ] − sin( ω t ) [ S xi S yi + S yi S xi ] (cid:111) . (4)In spin language, the pumping term in Eq. (2) thereforecorresponds to a time-dependent single ion anisotropywhose easy axis rotates with frequency ω around the z -axis. Of course, the magnon operators for YIG arenot directly related to Holstein-Primakoff bosons becausean additional Bogoliubov transformation is necessary todiagonalize the quadratic part of the boson hamiltionian.Nevertheless, we show in the appendix that also in thiscase the pumping term in the effective boson hamiltonian(2) can be related to a rotating easy axis anisotropy ofthe above type. III. KINETIC EQUATIONS
To discuss the time evolution of our toy model de-fined in Eq. (1) it is convenient to remove the explicittime dependence from the hamiltonian ˆ H ( t ) by perform-ing a canonical transformation to the “rotating referenceframe”, ˜ a = e i ω t a = ˆ U ( t ) a ˆ U † ( t ) , (5a)˜ a † = e − i ω t a † = ˆ U ( t ) a † ˆ U † ( t ) , (5b)where ˆ U ( t ) = e − i ω ta † a . The new operators satisfy theHeisenberg equations of motion i∂ t ˜ a = [˜ a, ˜ H ] , i∂ t ˜ a † = [˜ a † , ˜ H ] , (6)where the rotated hamiltonian ˜ H of our toy model doesnot depend explicitly on time,˜ H = ˜ (cid:15) ˜ a † ˜ a + γ a † ˜ a † + γ ∗ a ˜ a + u a † ˜ a † ˜ a ˜ a. (7)Here we have introduced the shifted oscillator energy˜ (cid:15) = (cid:15) − ω . (8)To relate correlation functions in the original model tothose in the rotating frame, we simply have to insert theappropriate phase factors. For example, in “S-theory”one usually considers the normal distribution function, n ( t ) = (cid:104) a † ( t ) a ( t ) (cid:105) = (cid:104) ˜ a † ( t )˜ a ( t ) (cid:105) , (9)and its anomalous counter-part, p ( t ) = (cid:104) a ( t ) a ( t ) (cid:105) = e − iω t (cid:104) ˜ a ( t )˜ a ( t ) (cid:105) ≡ e − iω t ˜ p ( t ) , (10)where expectation values are with respect to some densitymatrix ˆ ρ ( t ) specified at time t , (cid:104) . . . (cid:105) = Tr[ˆ ρ ( t ) . . . ] . (11)Throughout this work we shall mark all quantities definedin the rotating reference frame by a tilde. A. Instability of the non-interacting system
In the non-interacting limit ( u = 0) the equations ofmotion for the distribution functions n ( t ) and ˜ p ( t ) can beobtained trivially from the equations of motion (6) of theoperators ˜ a ( t ) and ˜ a † ( t ) in the rotating reference frame, i∂ t n ( t ) = γ ˜ p ∗ ( t ) − γ ∗ ˜ p ( t ) , (12a) i∂ t ˜ p ( t ) = 2˜ (cid:15) ˜ p ( t ) + γ [2 n ( t ) + 1] . (12b)These equations can be solved exactly. For | ˜ (cid:15) | > | γ | the solution is oscillatory, while in the strong pumpingregime | γ | > | ˜ (cid:15) | the solutions grow exponentially. Let us explicitly give the solution of Eqs. (12a,12b) with ini-tial conditions n (0) = n and ˜ p (0) = 0. For simplicity,we assume in the rest of this work that γ is real andpositive; the case of complex γ = | γ | e iϕ can be reducedto real γ > e iϕ into a re-definition of the anomalous correlator, e − iϕ ˜ p ( t ) → ˜ p ( t ).Defining α ≡ (cid:113) ˜ (cid:15) − γ , (13)the solution in the weak pumping regime γ < | ˜ (cid:15) | canbe written as Re˜ p ( t ) n + = − γ ˜ (cid:15) − cos(2 αt ) α , (14a)Im˜ p ( t ) n + = − γ sin(2 αt ) α , (14b) n ( t ) + n + = 1 + γ − cos(2 αt ) α . (14c)In the opposite strong pumping regime γ > | ˜ (cid:15) | the so-lution can be obtained by replacing α → iβ in the aboveexpressions, where β = (cid:113) γ − ˜ (cid:15) . (15)Then we obtainRe˜ p ( t ) n + = − γ ˜ (cid:15) cosh(2 βt ) − β , (16a)Im˜ p ( t ) n + = − γ sinh(2 βt ) β , (16b) n ( t ) + n + = 1 + γ cosh(2 βt ) − β . (16c)The behavior at the threshold value γ = | ˜ (cid:15) | can beobtained either from Eqs. (14a–14c) for α →
0, or fromEq. (16a–16c) for β → p ( t ) n + = − γ ˜ (cid:15) t , (17a)Im˜ p ( t ) n + = − γ t, (17b) n ( t ) + n + = 1 + 2 γ t . (17c)Physically, the exponential increase of correlations for γ > | ˜ (cid:15) | is a consequence of the fact that in this regimethe non-interacting part of the hamiltonian ˜ H in Eq. (7)is not bounded from below. This is easily seen by setting˜ a = ˆ X + i ˆ P √ , ˜ a † = ˆ X − i ˆ P √ , (18)so that˜ (cid:15) ˜ a † ˜ a + γ a † ˜ a † + ˜ a ˜ a ] = ˜ (cid:15) − γ P + ˜ (cid:15) + γ X . (19)Obviously, for γ > | ˜ (cid:15) | the non-interacting part of ourtoy model describes a harmonic oscillator with negativemass. The spectrum of such a quantum mechanical sys-tem is not bounded from below, which gives rise to theexponential growth of correlations discussed above. For-tunately, this pathology of the non-interacting limit iscured for any positive value of the interaction. The phys-ical consequences of this are most transparent if we con-sider the equations of motion for the expectation valuesof the creation and annihilation operators, which will bediscussed in the following subsection. B. Gross-Pitaevskii equation
The toy model hamiltonian (7) in the rotating refer-ence frame gives rise to the following Heisenberg equationof motion for the annihilation operator, i∂ t ˜ a = ˜ (cid:15) ˜ a + γ ˜ a † + u ˜ a † ˜ a . (20)Taking the expectation value of both sides and factorizingthe expectation value of the interaction term as follows, (cid:104) ˜ a † ˜ a (cid:105) → (cid:104) ˜ a † (cid:105)(cid:104) ˜ a (cid:105) , (21)we obtain the Gross-Pitaevskii equation for the time-dependent order-parameter φ ( t ) ≡ (cid:104) ˜ a ( t ) (cid:105) in the rotatingreference frame, i∂ t φ = ˜ (cid:15) φ + γ φ ∗ + u | φ | φ = ∂H cl ( φ ∗ , φ ) ∂φ ∗ , (22)where the effective classical hamiltonian H cl is given by H cl ( φ ∗ , φ ) = ˜ (cid:15) | φ | + γ φ ∗ + φ ] + u | φ | . (23)Writing φ = ( X + iP ) / √ H cl ( X, P ) = ˜ (cid:15) − γ P + ˜ (cid:15) + γ X + u X + P ) . (24)Because the classical hamiltonian H cl ( X ( t ) , P ( t )) is con-served along the flow defined by the Gross-Pitaevskiiequation, the solutions of Eq. (22) are simply given by thecurves of constant H cl ( X ( t ) , P ( t )) in phase space. Theshape of H cl and typical trajectories are shown in Fig. 1.Note that in the strong pumping regime γ > | ˜ (cid:15) | thefunction H cl ( X, P ) has two degenerate minima at X = 0 , P = ± P ∗ = ± (cid:114) γ − ˜ (cid:15) ) u , (25)corresponding to stationary points (in the rotating ref-erence frame) of the system. Note that at these specialpoints the expectation value of the annihilation operatoris purely imaginary, (cid:104) ˜ a (cid:105) = ± i √ P ∗ = ± i (cid:114) γ − ˜ (cid:15) u . (26) FIG. 1: (Color online) Graph of the classical hamiltonian H cl ( X, P ) defined in Eq. (24). The corresponding classicalhamiltonian equations of motion are equivalent to the Gross-Pitaevskii equation (22) for the complex order parameter φ ( t ) = ( X ( t ) + iP ( t )) / √
2. The thick black lines are solutionsof the equations of motion for different initial conditions. X and P are both measured in units of the momentum scale | P ∗ | = (cid:112) | γ − ˜ (cid:15) | /u . (a): ˜ (cid:15) /u = 10 and γ /u = 2; note thatfor | ˜ (cid:15) | > γ our classical hamiltonian H cl ( X, P ) has a globalminimum for X = P = 0. (b): ˜ (cid:15) /u = 10 and γ /u = 40;in the regime γ > | ˜ (cid:15) | our classical hamiltonian has two de-generate minima at ( X, P ) = (0 , ± P ∗ ), so that the graph of H cl ( X, P ) shown in (b) has some similarity to the shape ofNapoleon’s hat. The associated stationary points of the dynamical system(22) describe a coherent magnon state where the macro-scopic magnetization has a rotating component perpen-dicular to the static magnetic field. In bosonic language,such a state corresponds to a coherent state, which isan eigenstate of the annihilation operator.
Whetheror not this state should be called a Bose-Einstein con-densate of magnons seems to be a semantic question. Inour opinion this terminology is somewhat misleading, be-cause this coherent magnon state does not exhibit spon-taneous symmetry breaking which is one of the most im-portant properties of a Bose-Einstein condensate in in-teracting Bose gases. Instead, the coherent magnon stateobserved by Demokritov and co-workers is generatedby an external pumping field which explicitly breaks the U (1)-symmetry of the magnon hamiltonian. In the staticlimit, the role of a similar symmetry breaking term onthe Bose-Einstein condensation of magnons has recentlybeen discussed by Dell’Amore, Schilling, and Kr¨amer. C. Time-dependent Hartree-Fock approximation
Let us now take into account the leading fluctuationcorrection to the replacement (21) in the derivation ofthe Gross-Pitaevskii equation (22). To first order in u ,fluctuations simply renormalize the bare parameters ˜ (cid:15) and γ in Eq. (22) as follows,˜ (cid:15) → ˜ (cid:15) c ( t ) = ˜ (cid:15) + 2 un c ( t ) , (27a) γ → γ c ( t ) = γ + u ˜ p c ( t ) , (27b)where the connected correlation functions n c ( t ) and ˜ p c ( t )in the rotating reference frame are defined by n c ( t ) = (cid:104) δ ˜ a † ( t ) δ ˜ a ( t ) (cid:105) , (28a)˜ p c ( t ) = (cid:104) δ ˜ a ( t ) δ ˜ a ( t ) (cid:105) , (28b)with δ ˜ a ( t ) = ˜ a ( t ) −(cid:104) ˜ a ( t ) (cid:105) . Instead of the Gross-Pitaevskiiequation (22) we now obtain for the order parameter dy-namics, i∂ t φ = ˜ (cid:15) c ( t ) φ + γ c ( t ) φ ∗ + u | φ | φ. (29)Note that this generalized Gross-Pitaevskii equation de-pends on the connected correlation functions n c ( t ) and˜ p c ( t ), which we calculate in self-consistent Hartree-Fockapproximation. The resulting equations of motion can beobtained from the corresponding non-interacting kineticequations (12a,12b) by substituting˜ (cid:15) → ˜ (cid:15) ( t ) = ˜ (cid:15) + 2 u [ n c ( t ) + | φ ( t ) | ] , (30a) γ → γ ( t ) = γ + u [˜ p c ( t ) + φ ( t )] . (30b)The kinetic equations for the connected distribution func-tions are therefore i∂ t n c ( t ) = γ ( t )˜ p ∗ c ( t ) − γ ∗ ( t )˜ p c ( t ) , (31a) i∂ t ˜ p c ( t ) = 2˜ (cid:15) ( t )˜ p c ( t ) + γ ( t )[2 n c ( t ) + 1] . (31b)For φ = 0 these equations reduce to the kinetic equationsobtained within “S-theory”. The numerical solution ofEqs. (29, 31a, 31b) for n c (0) = n , ˜ p c (0) = 0, and in-finitesimal Im φ ( t ) > φ (0) builds up to a finite oscillation. Moreover,the connected correlation functions n c ( t ) and ˜ p c ( t ) re-main always bounded, in contrast to the exponentiallygrowing correlations in the non-interacting limit given inEqs. (16a–16c). Note also that the time evolution of theconnected correlation functions appears to be rather ir-regular as soon as the order-parameter has built up toa finite value. In the conventional “S-theory” the quan-tities n c and ˜ p c are periodic (Fig. 2c), while includingthe order parameter dynamics disturbes this strict peri-odicity (Fig. 2b). This feature is still missing within theusual “S-theory”in the strong pumping regime. −1 0 1 2 3 0 10 20 30 40 50 a t (a) n c ( t ) / n Im ~ p c ( t ) / n Im f ( t ) / | P * | −2 0 2 4 6 0 10 20 30 40 50 b t (b) 10 −3 n c ( t ) / n −3 Im ~ p c ( t ) / n Im f ( t ) / | P * | −2 0 2 4 6 0 10 20 30 40 50 b t (c) 10 −3 n c ( t ) / n −3 Im ~ p c ( t ) / n FIG. 2: (Color online) Numerical solution of the coupled ki-netic equations (29, 31a, 31b) with initial conditions n c (0) = n = 1, ˜ p c (0) = 0, and φ (0) = 0 . i . The character-istic energy scales α and β are defined in Eqs. (13, 15).(a): ˜ (cid:15) /u = 500 and γ /u = 200. Recall that in the absenceof interactions there is no instability as long as | ˜ (cid:15) | > γ .(b): ˜ (cid:15) /u = 500 and γ /u = 5000. In this regime there wouldbe an instability in the non-interacting limit, but in the in-teracting system all correlations remain finite. (c): Same pa-rameters as (b) but without finite expectation values, like inthe conventional “S-theory”. IV. SUMMARY AND CONCLUSIONS
Let us briefly summarize the two main results of thiswork:First of all, we have shown that a complete theoreti-cal description of the coherent magnon state emerging inYIG for sufficiently strong microwave pumping requiresan extension of the usual “S-theory” which includes theGross-Pitaevskii type of equation for the expectation val-ues of the magnon operators. Within a simple toy modelconsisting only of a single magnon mode we have shownhow to construct such an extension. The explicit solutionof the resulting kinetic equations shows that the orderparameter dynamics strongly influences the distributionfunctions.Our second main result is the observation that inspin-language the usual bosonic resonance hamiltonian(2) corresponds to a time-dependent rotating easy axisanisotropy whose axis is perpendicular to the direction ofthe external field. If this anisotropy is sufficiently strong,it gives rise to a forced oscillation of the macroscopicmagnetization around the direction of the static exter-nal field. Although this phenomenon can be described interms of a coherent magnon state, it should not be calleda Bose-Einstein condensate, because the emergence ofthis state is not associated with any kind of spontaneoussymmetry breaking.In future work, we shall further extend our approachin two directions: on the one hand, a realistic model forYIG involves a quasi-continuum of magnon modes, whichcan condense at finite wave-vectors ± k ∗ . For a morerealistic quantitative description of the experiments, weshould therefore generalize our extended “S-theory” toinclude all magnon modes relevant to the experimentson YIG. This would also allow us to distinguish betweenthe “primary magnons” created by the external pump-ing, and the “condensing magnons” with wave-vectors atthe minima of the dispersion. The second direction forimproving our approach is to include correlation effectsbeyond the self-consistent Hartree-Fock approximationinto the kinetic equations. For example, to second orderin u the kinetic equations will contain relaxation termswhich will damp the oscillatory time dependence foundat the Hartree-Fock level. Work in both directions is inprogress. ACKNOWLEDGMENTS
We thank L. Bartosch and A. Isidori for discussionsand acknowledge financial support by SFB/TRR49 andthe DAAD/CAPES PROBRAL-program. The work byA. K. and P. K. was partially carried out at the Interna-tional Center for Condensed Matter Physics at the Uni-versity of Bras´ılia, Brazil. We thank A. Ferraz for hishospitality. We are grateful to Y. M. Bunkov and G. E.Volovik for drawing our attention to analogies betweenexperiments on YIG and superfluid He, and to A. A.Zvyagin for sending us a copy of Ref. [24].
APPENDIX: PARALLEL PUMPING OFMAGNONS IN YIG
It is generally accepted that the magnetic proper-ties of YIG in the parallel pumping geometry can bemodelled by the following time-dependent quantum spin model, ˆ H YIG ( t ) = − (cid:88) ij (cid:88) αβ (cid:104) J ij δ αβ + D αβij (cid:105) S αi S βj − [ h + h cos( ω t )] (cid:88) i S zi , (A1)where α, β = x, y, z label the three spin components, andthe exchange couplings J ij = J ( r i − r j ) are only finite ifthe lattice sites r i and r j are nearest neighbors on a cubiclattice with lattice spacing a ≈ . J ≈ . D αβij = D αβ ( r i − r j ) is explicitly D αβij = (1 − δ ij ) µ | r ij | (cid:104) r αij ˆ r βij − δ αβ (cid:105) , (A2)where r ij = r i − r j and ˆ r ij = r ij / | r ij | . If we arbitrar-ily set the magnetic moment µ = 2 µ B = e (cid:126) / ( mc ), thenwe should work with an effective spin S ≈ .
2, as dis-cussed in Ref. [21]. Here h and h are the amplitudesof the static and oscillating magnetic field (multiplied by µ ). We assume that h > | h | and that both the staticand the oscillating magnetic field point into the directionof the macroscopic magnetization which we call the z -axis. At this point one might already wonder how in thisparallel pumping geometry one can possibly arrive at abosonic resonance hamiltonian of the form (2), which ac-cording to Eq. (4) can be related to some rotating easyaxis anisotropy. In fact, we shall show shortly that thespin hamiltonian (A1) with parallel pumping cannot bereduced to the bosonic resonance hamiltonian (2).To bosonize the hamiltonian (A1) we express the spinoperators in terms of boson operators b i and b † i by meansHolstein-Primakoff transformation, S + i = √ S (cid:115) − b † i b i S b i = ( S − i ) † , (A3a) S zi = S − b † i b i . (A3b)As usual, the square roots are then expanded in powersof 1 /S , resulting in a hamiltonian of the formˆ H YIG ( t ) = H ( t ) + ˆ H ( t ) + ˆ H int , (A4)where H ( t ) is a time-dependent constant, ˆ H ( t ) isquadratic in the boson operators, and the time-independent interaction ˆ H int involves three and more bo-son operators. After Fourier transformation to momen-tum space the quadratic part of the hamiltonian can bewritten asˆ H ( t ) = (cid:88) k (cid:20) A k b † k b k + B k b † k b †− k + B ∗ k b − k b k (cid:21) + h cos( ω t ) (cid:88) k b † k b k . (A5)where A k = A − k = (cid:88) i e − i k · r ij A ij , (A6a) B k = B − k = (cid:88) i e − i k · r ij B ij , (A6b)with A ij = δ ij h + S ( δ ij (cid:88) n J in − J ij )+ S (cid:34) δ ij (cid:88) n D zzin − D xxij + D yyij (cid:35) , (A7a) B ij = − S (cid:2) D xxij + 2 iD xyij − D yyij (cid:3) . (A7b)Finally, we use a Bogoliubov transformation to diagonal-ize the time-independent part of ˆ H ( t ), (cid:18) b k b †− k (cid:19) = (cid:18) u k − v k − v ∗ k u k (cid:19) (cid:18) a k a †− k (cid:19) , (A8)where u k = (cid:114) A k + (cid:15) k (cid:15) k , v k = B k | B k | (cid:114) A k − (cid:15) k (cid:15) k , (A9)and (cid:15) k = (cid:113) A k − | B k | . (A10)After this transformation the hamiltonian reads ˆ H ( t ) = (cid:88) k (cid:20) (cid:15) k a † k a k + (cid:15) k − A k (cid:21) + h cos( ω t ) (cid:88) k (cid:20) A k (cid:15) k a † k a k + A k − (cid:15) k (cid:15) k (cid:21) + (cid:88) k (cid:104) γ k cos( ω t ) a † k a †− k + γ ∗ k cos( ω t ) a − k a k (cid:105) , (A11)where γ k = − h B k (cid:15) k . (A12)To obtain the quadratic part of the resonance hamilto-nian (2) from Eq. (A11) two additional approximationsare necessary: the second line on Eq. (A11) involving thecombination cos( ω t ) A k a † k a k has to be dropped, while inthe last line one should substitute γ k cos( ω t ) → γ k e − iω t , γ ∗ k cos( ω t ) → γ ∗ k e iω t . )(A13)Apparently this approximation has been accepted formany decades in the literature. However, a thoroughstudy of the non-resonant terms neglected in this ap-proximation has been performed by Zvyagin et al. , who showed that the neglected terms can qualitatively changethe results obtained in resonance approximation. Herewe would like to point out that the approximations lead-ing to Eq. (A13) amount to an essential modification ofthe original spin hamiltonian. To see this, let us for themoment accept the validity of these approximations, thusreplacing Eq. (A11) by the non-interacting part of theresonant hamiltonian (2),ˆ H ( t ) ≈ (cid:88) k (cid:15) k a † k a k + 12 (cid:88) k (cid:104) γ k e − iω t a † k a †− k + γ ∗ k e iω t a − k a k (cid:105) , (A14)where we have dropped the constant terms. Using nowthe inverse of the Bogoliubov transformation (A8) to re-express the magnon operators in Eq. (A14) in terms ofHolstein-Primakoff bosons and assuming for simplicitythat γ k is real, the second term in Eq. (A14) can bewritten as 12 (cid:88) k (cid:104) γ k e − iω t a † k a †− k + γ k e iω t a − k a k (cid:105) = 12 (cid:88) k (cid:26) γ k A k (cid:15) k cos( ω t ) (cid:104) b † k b †− k + b − k b k (cid:105) + iγ k sin( ω t ) (cid:104) b † k b †− k − b − k b k (cid:105)(cid:27) + (cid:88) k γ k B k (cid:15) k cos( ω t ) (cid:20) b † k b k + 12 (cid:21) . (A15)Only the last term on the right-hand side has the form ofthe boson representation of the Zeemann term associatedwith an external pumping field parallel to the magneti-zation, while the first two terms can be identified withthe boson representation of spin anisotropies associatedwith a rotating easy axis perpendicular to the z -axis,see Eq. (4). We thus conclude that the time-dependentpart of the resonant hamiltonian (2) does not representthe time-dependent Zeemann energy associated with aharmonically oscillating magnetic field in the directionof the magnetization. Instead, the time-dependent off-diagonal pumping terms arise from a rotating easy axisanisotropy perpendicular to the magnetization. The mi-croscopic origin of such a term is not clear to us; pos-sibly the time-dependent electric field associated withthe harmonically varying magnetic field parallel to themagnetization can indirectly induce such a term in thespin hamiltonian, similar to the second order interactionhamiltonian in the theory of two-magnon Raman scatter-ing in antiferromagnets. Moreover, in real materialscrystallographic or shape anisotropies can give rise to fur-ther contributions to the effective spin hamiltonian whichafter Holstein-Primakoff transformation might have thesame form as the terms in Eq. (A15). V. Cherepanov, I. Kolokolov, and V. S. 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