Table Of Content

The open Heisenberg chain under boundary fields: a magnonic logic gate Gabriel T. Landi∗ Universidade Federal do ABC, 09210-580 Santo André, Brazil Dragi Karevski Institut Jean Lamour, Department P2M, Groupe de Physique Statistique, Université de Lorraine, CNRS, B.P. 70239, F-54506 Vandoeuvre les Nancy Cedex, France (Dated: June30, 2015) WestudythespintransportinthequantumHeisenbergspinchainsubjecttoboundarymagnetic fields and driven out of equilibrium by Lindblad dissipators. An exact solution is given in terms of 5 matrix product states, which allows us to calculate exactly the spin current for any chain size. It 1 is found that the system undergoes a discontinuous spin-valve-like quantum phase transition from 0 ballistic to sub-diffusive spin current, depending on the value of the boundary fields. Thus, the 2 chain behaves as an extremely sensitive magnonic logic gate operating with the boundary fields as thebase element. n u J I. INTRODUCTION dynamical equation.31,32 However, these models, being 9 quantummany-bodyproblems,canseldombe solvedex- 2 actlyandfromanumericalpointofviewtheycanusually One of the fundamental issues in condensed matter ] physics is the determination of macroscopic parameters only be solved for small lattices. h fromthe underlying microscopic properties. For systems The purpose of this paper is to study the transport p in equilibrium, the Gibbsian approach gives an elegant properties in the NESS of the one-dimensional Heisen- t- solution since it depends only on the underlying micro- berg chain coupled to two Lindblad reservoirs at each n scopic energy spectrum. However, even if substantial end, and also subject to magnetic fields at its bound- a progress has recently been made in understanding non- aries. Remarkably, the steady state of this model is ex- u q equilibrium systems, in particular through the so called actly expressible in terms of a matrix product state26,27 [ fluctuation theorems,1–5 no such approach is available involving operators satisfying the SU(2) algebra (in the for systems in a Non-Equilibrium Steady-State (NESS), case of an XXZ chain this generalizes to the quantum v2 characterized by the existence of steady currents. This Uq[SU(2)] algebra27). This provides a method to com- 2 forces one to resort to a full dynamical calculation in putethesteady-statespincurrentJ forany chain size.28 3 order to extract steady-state parameters. Such a diffi- We will show that depending on the strength of the ap- 7 cultyisinherentofnon-equilibriumsystems,datingback pliedmagneticfield,J mayundergoadiscontinuousspin- 7 to Drude’s calculation of the electrical conductivity of valve-likequantumphasetransitionfromballistictosub- 0 metals in 1900.6 As another example, we note the re- diffusive(J ∼1/N2;cf. Fig.3(d)below). Asweshalldis- 1. centdiscussionsconcerningthe microscopicderivationof cuss,theoriginofthistransitionisrelatedtotheentrap- 0 Fourier’s law in insulating crystals.7–11 ment of magnons inside the chain caused by the bound- 5 A more thorough understanding of the NESS is also aryfields which, inturn, increasethe number ofmagnon 1 essential for the development of several applications in scatteringevents. Wearguethatoursystemmaybeused : v phononics,12–14 spintronics,15–18 andmagnonics.19,20 We as an extremely sensitive magnonic logic gate operating i point in particular to two recent remarkable papers by with an external magnetic field as the base element. X Chumaket.al.20andOltscheret.al.18. InRef.20theau- r a thorsreportonamagnoniclogicgate,wherethemagnon currentis adjusted by controllingthe number ofmagnon II. DESCRIPTION OF THE MODEL scattering processes induced by an auxiliary magnon in- jector (the base). On a different setting the authors in We consider the isotropic Heisenberg spin-1/2 chain Ref. 18 study the transport of spin polarized current in with N sites described by the Hamiltonian a two dimensional electron gas. They observe for the first time the existence of a ballistic spin flow, in stark N−1 1 disagreement with classical predictions. H = σxσx +σyσy +σzσz +h(σz −σv ), The transport properties reported in Refs. 18 and 20 2 i i+1 i i+1 i i+1 1 N i=1 both involve the presence of a NESS. Moreover, they X (cid:0) (cid:1) (1) share in common the fact that they cannot be explained where the σ’s are the usual Pauli matrices. The last byclassicaltheories,thusrequiringafullquantumtreat- termdescribestheZeemaninteractionexperiencedbythe ment. On the theoretical side, these quantum NESS boundaryspinswithafieldpointinginthez directionon are usually implemented on 1d lattice spin systems cou- thefirstsiteandinthe−n =(sinθ,0,−cosθ)direction v pledtoexternalreservoirs.14,21–30 Theeffectofthereser- on the last site. Note that with this parametrizationthe voirs is quite often described by a non-unitary Lindblad boundary fields point in opposite directions when θ =0. 2 Thechainiscoupledtotworeservoirsateachendsuch first step is to note that since ρ is a Hermitian positive that its density matrix ρ is governed by the Lindblad semi-definite operator, we may use the following param- master equation32 eterization: SS† dρ ρ= . (5) =−i[H,ρ]+DL(ρ)+DR(ρ), (2) tr(SS†) dt ForaHeisenbergchainmadeofN spins1/2,theoperator where the left and right dissipators D are given by L(R) S lives on the Hilbert space H=C2N. We now use the ansatz that S can be described by a D (ρ)= 2KαρKα†−{Kα†Kα,ρ}, (3) α r r r r matrix-product state: r=± X S =hφ|Ω⊗N|ψi (6) withKL = γ(1±f)σ± andKR = γ(1∓f)σv ± and ± 1 ± N where σv ± are the ladder operators in the n direction. where Ω is a 2×2 matrix with operator-valued entries fEoxrptlhiceitllNoywoepnrienghaospeσrNva−tor=an(cdosthθeσNxad−pjoiinσtNye+xpvsrienssθiσoNzn)f/o2r Ω= Sz S+ =S σz +S σ++S σ− . (7) z + − the raising one. S− −Sz! The forcing term f ∈ [0,1] describes the polarization The operators S live in an auxiliary space A so that a of the spinreservoirsand is relatedto a reservoirinverse Ω⊗N ∈ H⊗A. After contracting with |φi and |ψi we temperature β by f = tanh(β). At f = 1 (zero tem- recover S ∈ H. From the bulk structure of the Hamilto- perature) the left bath corresponds to a perfect magnon nian(1),itcanbeshownthatifEq.(6)istobeasolution, source, pumping magnons into the system at a rate γ, then the operators S must obey the SU(2) algebra: a while the right dissipator is a perfect drain, absorbing magnons at the same rate. We shall concentrate mostly [Sz,S±]=±S± (8) onf =1,eventhoughsomewordswillbegivenforf <1. [S ,S ]=2S . (9) Note that when h>0(<0)the boundary fields point in + − z the same (opposite) direction as the dissipators (Fig. 1). The proper representation of the algebra to be explicitly used in the MPS solution is specified by a complex representation parameter p which is fixed by substitut- ing Eq. (6) and (5) into the steady-state equation (4) and solving the resulting equations. It turns out that it is fixed by a lowest weight condition: hφ|S = phφ|, z where hφ| ≡ h0| is a lowest weight state of the representation. Explicitly,intermsofasemi-infinite setofstates {|ni}∞ one has the irreducible representations n=0 ∞ S = (p−n)|nihn| (10) z n=0 X ∞ S = (n+1)|nihn+1| (11) FIG. 1. (Color Online) Schematic drawing of the dissipators + DL,R(ρ)(black,solid arrows) andtheboundaryfieldsh(red, nX=0 dashedarrows) actingonthefirstandlast spinsofthechain. ∞ In (a) the fields are in the same direction as the dissipators S = (2p−n)|n+1ihn|. (12) − (h>0) and in (b) the fields act in directions opposite to the n=0 dissipators (h<0). X Notice that for half-integer values of p these representations reduce to the usual finite dimensional representations of SU(2). In the present case, the representation III. OUTLINE OF THE MATRIX PRODUCT parameter turns out to be STATE SOLUTION i p= (13) 2(γ−ih) TheuniqueNESSattainedbythesystematlongtimes which fixes the associated infinite dimensional represen- is the solution of Eq. (2) with dρ/dt=0: tation of SU(2). The right state |ψi over which Ω⊗N is i[H,ρ]=D (ρ)+D (ρ). (4) evaluated is given by the coherent state L R ∞ 2p Atf =1theexactsolutionwasfoundinRef.27interms |ψi= ψn |ni, ψ =−tan(θ/2). (14) ofamatrixproductstate(MPA),aswenowoutline. The n n=0 (cid:18) (cid:19) X 3 Including these results in Eqs. (6) and (5) gives a com- where Z(N) is the normalization constant, plete solution for the density matrix of the steady-state. From this general solution it is possible to compute Z(N)=tr(ρ)=h0,0|BN|ψ,ψ∗i. (20) 0 the expectationvalueofanylocalobservable,28 the most important of which is the spin current J leaving site The explicit computation of Z(N) requires constructing i i toward site i + 1. It is defined from the continuity the matrix equation (B ) =2|p−k|2δ +ℓ2δ +|2p−ℓ|2δ . 0 k,ℓ k,ℓ k,ℓ−1 k,ℓ+1 dhσzi i =Ji−1−Ji We then have dt where N 2p 2 Z(N)= (BN) tan2k(θ/2) . 0 0,k k Ji =hσixσiy+1−σiyσix+1i Xk=0 (cid:12)(cid:12)(cid:18) (cid:19)(cid:12)(cid:12) (cid:12) (cid:12) These equations are valid for i = 2,...,N −1. Slightly In particular, when θ = 0 one has (cid:12)simply(cid:12) Z(N) = differentequationsapplytotheboundaries. Inthesteady (B0N)0,0. state dhσzi/dt=0 which gives It can be further shown that i J1 =J2 =...=JN :=J [Bx,By]=2i(Tz−Sz)B0 The expectation value of an arbitrary observable A and that T −S commutes with B . This reflects the z z 0 may be computed as translational symmetry of the J in the steady-state. i Hence, making use of Eq. (10), we arrive at tr(S†AS) hAi=tr(Aρ)= tr(S†S) 2γ Z(N −1) J = . (21) γ2+h2 Z(N) Our strategy will be to first trace over the Hilbert space and write everything in terms of expectation values on This is the required formula for the steady-state magne- the auxiliary space. But note that S and S† will each tization flux. In the present model J is a function of N, contain an auxiliary space. So when we write SS† we h, γ and θ only. Eq. (21) must be computed for each N. must double our auxiliary space. That is, we write Even though this may be done exactly, the formulas be- comeextremelycumbersomeforlargesizes. Ontheother SS† =h0,0|Ω(p)ΩT(p∗)|ψ,ψ∗i hand, computing J numerically is now a trivial task. Also of interest is the magnon density hn i = (1 + where Ω(p) and ΩT(p∗) act ondifferent auxiliaryspaces. i hσzi)/2. A calculation similar to the above leads to Moreover, |ψ∗i is defined as |ψi in Eq. (14), but with p∗ i insteadofp. Similarly,ΩT(p∗)isdefinedinawaysimilar h0,0|Bi−1B BN−i|ψ,ψ∗i to Eq. (7): hσzi= 0 z 0 (22) i Z(N) ΩT(p∗):=T σz +T σ−+T σ+ z + − where the operators T are defined with p∗ instead of p. IV. RESULTS a Moreover, they commute with the S since they act on a different auxiliary spaces. We now discuss the behavior of J as a function of N, Next define h,γ andθ. Thefocuswillbeonthecasef =1,forwhich the MPSsolutionisvalid. Notwithstanding,afew words B =tr[σaΩ(p)ΩT(p∗)], a∈{0,x,y,z}. (15) a will be given about the case f <1. We begin with θ = 0 and h = 0. The spin current as Explicitly we have a function of N and γ is presented in Fig. 2. In order to interpret these results, recall that magnons are con- B =2S T +S T +S T (16) 0 z z + + − − stantlybeingpumpedattheleftsource,whichthenprop- B =(S −S )T +S (T −T ) (17) agate through the lattice and are eventually collected in x − + z z − + the right drain. The spin current is then simply pro- By =i[Sz(T−+T+)−(S−+S+)Tz] (18) portional to the number of magnons being collected at the right drain. This number depends on two things: (i) B =S T −S T . (19) z + + − − the number of magnons being injected per unit time in the left source, which is proportional to γ and (ii) the The spin current may then be written as magnon scattering events during the trip to the right 1 drain. In standard electrical conduction (e.g. in Drude’s J = h0,0|Bi−1[B ,B ]BN−i−1|ψ,ψ∗i i Z(N) 0 x y 0 model), the electrons scatter with lattice imperfections 4 FIG. 2. (Color Online) Spin current J for f = 1, θ = 0 and h = 0. (a) J vs. γ for different sizes N. (b) J vs. N for different values of γ. The dotted black line has slope -2. orphonons. Since the numberofscatteringagentsscales proportionally to N we then have a diffusive current J ∼ 1/N. In our case the magnons do not scatter with lattice imperfections. They either travel through unim- peded or they participate in 4-magnon scattering events (where 2 magnons scatter producing two new magnons in the process33). When γ is sufficiently small the density of magnons in the chain is very small, thus making FIG. 3. (Color Online) Spin current J as a function of the theseeventsveryrare. InthiscaseJ willincreasewithγ boundary fields h with f = 1 and θ = 0. (a) N2J vs. h for and will also be independent of N; i.e., ballistic. This is γ = 1 and different values of N. (b) J/γ vs. h for N = 100 ∗ and different values of γ around γ = 1/N = 0.01. (c) and clearlyobservedinFig.2(a),whereweseethatthecurves (d) J/γ vs. h for γ = 10−5 and different values of N. The for different N overlap when γ is small. Conversely, in dashed lines in (d) correspond to Eq. (24). the high γ limit the number of magnons, and hence the number of scattering events, will be significant. In this regime it is found28 that J is sub-diffusive, behaving as J ∼1/N2. Thereasonforthisisthatbydoublingthesize structure at γ >γ∗ to a plateau at γ <γ∗. This plateau of the chain, we quadruple the number of four-magnon is illustrated in more detail in Figs. 3(c) and (d) for scattering events. As shown in Ref. 28 the transition γ =10−5 anddifferentsizes. As canbe seen,the plateau between the ballistic and sub-diffusive regimes occurs at region is asymmetric with respect to h and independent ofsize. Itcorrespondstotheballisticbehaviorofthespin 1 γ∗ ≃ (23) current. As the field is increased, however, one eventu- N ally observes an abrupt transition to a much lower spin current. For positive fields the transition is continuous Aclearexampleofthis transitionis seeninthe curvefor γ =10−2inFig.2(b),wheretheregimechangesabruptly whereas for negative fields it is discontinuous (strictly from J ∼1 to J ∼N−2 exactly at N =100. speaking, it is only discontinuous in the thermodynamic limit). The criticalfield where the plateau transitionoc- Nextwediscussthebehaviorfornon-vanishingbound- cursisfoundfromthe simulationstobe h∗ ≃−5/N. We ary fields, h 6= 0, still keeping θ = 0. In Fig. 3(a) we present N2J vs. h for γ =1 (sub-diffusive; high magnon alsocallattentiontothefactthatoutsidetheplateaure- gion, J is againwelldescribedby Eq.(24), asillustrated density). Ascanbeseen,evenformoderatelysmallsizes, thecurvesstarttoscaleverywellaccordingtoJ ∼1/N2. by the dashed lines in Fig. 3(d). This indicates that for large fields the behavior is again sub-diffusive. In this scaling region we have found that the current is very well described by The results presented so far were obtained from the exactMPAsteadystatewhichisvalidonlyatf =1(zero π2 1 temperature). However, the rich behavior of the current J ≃ , (24) γN21+ 2h + h2 observed for f = 1 also survives at finite temperatures; γ2N γ2 i.e.,forf <1. ThiscanbeseeninFig.4wherewereport which is illustratedby the dashedline in Fig. 3(a). Note the currentJ vs.f asobtainedfromthe exactnumerical alsothatJ isasymmetricwithrespecttoh; i.e.,thespin diagonalization14 of Eq. (2) for N = 6. The current as current is rectified.14 seenfromthenumericsshowsbasicallythesamefeatures The changes which occur as we reduce γ below γ∗ asintheMPAcase: abell-shapedbehaviorathighγ and are illustrated in Fig. 3(b), where we plot J/γ vs. h a sharp plateau at low γ (for this small size the plateau for N = 100 and different values of γ. As can be seen, is not yet completely formed). In Fig. 4 we also plot thereisadrasticbehavioraltransitionfromabell-shaped the MPS solution when f = 1 to illustrate the perfect 5 FIG. 4. (Color Online) Spin current J vs. h when f < 1, computed using theexact diagonalization of Eq. (2) for N = 6. (a)γ =10−5. (b)γ =1. Thedottedblacklinescorrespond to theMPS solution when f =1. FIG. 6. (Color Online) J vs. h for N = 500 and different valuesofθ(asdefinedinFig.1). (a)γ =10−4 and(b)γ =1. FIG. 5. (Color Online) Small size effects in the spin current. (a) J/γ vs. Nh for γ = 10−5 and different values of N. (b) J/γ vs. Nh for N =15 and different values of γ. agreement between both methods. Thegradualformationofthe plateauasthe sizeofthe system increases in illustrated in Fig. 5(a). In Fig. 5(b) we showthe changeswhichoccur asone changesγ when N =15. As can be seen in both images and in Fig. 3(c), whenN issmallthe currentpresentsaseriesofirregular and sharp resonances when h < 0, at positions which vary with N (such peaks have been observed recently in FIG. 7. (Color Online) Magnon density profile hnii = (1+ Ref. 34). It is important to note, however, that these hσzi)/2fordifferentvaluesofh,withN =500,γ =10−5 and i peaks only appear for γ ≤ 1/N2 and therefore become θ = 0. (a) h < 0 near the plateau transition [cf. Fig. 3(d)]. vanishingly small for any moderately large size. This (b) h>0. can be seen, for instance, by comparing the curves with γ =10−4 andγ =10−2inFig.5(b). Botharepractically V. DISCUSSION AND CONCLUSIONS identical, except for the peaks, which are only present when γ = 10−4. Note also that it follows from Eq. (21) that J is bounded so that these cannot be delta peaks. The remarkable and sharp transitions observed in the Weconsidernowthecasewithageneraltwistingangle spin current,from ballistic (inside) to sub-diffusive (out- θ ∈ [0,π]. Fig. 6(a) shows J vs. h for different values of side the plateau), as the magnitude |h| of the boundary θ with fixed size N = 500 and γ = 10−4. As expected, fieldsisincreased,suggestthatsufficientlyhighfieldsact J → 0 as θ → π. However, and remarkably, even for asscatteringbarriers,impedingmagnonstoflowthrough values of θ close to the undriven situation θ = π, one thesystem,fromsourcetodrain. Thiscanalsobeseenby still observes high values of J for negative values of h, lookingatthemagnondensityprofilehn i=(1+hσzi)/2 i i in a plateau region that shrinks as θ → π. Thus, by plotted in Fig. 7 for N =500, γ =10−5 and θ =0. The monitoring the twisting angle θ, one can fine-tune the red (solid) curve in Fig. 7(a) corresponds to the profile high current plateau width. For completeness, we also intheplateau(ballistic)regionofFig.3(d). Inthis case, show the behavior for large γ in Fig. 6(b). the distribution is flat with hn i ≃ 1/2, characteristic of i 6 a maximalcurrentstate. Onthe other hand,outside the application of them as describing the injection and col- plateau the profile is sine-shaped, characteristic of the lection of magnons. sub-diffusive regime.26 The transition between the two The energy and time units of the problem are set by profiles is discontinuous for h<0 [Fig. 7(a)] and contin- the constant J which should appear in the first term of uous for h > 0 [Fig. 7(b)]. Hence, we conclude that the Eq. (1), but which we have throughout set as unity. Ac- densityofmagnonsinsidethechainmayalsobeadjusted cording to Ref. 35, J ∼ 10−22 J. The pumping rate γ bychangingtheboundaryfieldh. Chumaket. al.20 used (measuredin magnonsper second)shouldoperate below asimilarideatoconstructtheirmagnoniclogicgate. But the critical value γ∗ which, in the correct units, reads intheircaseanadditionalsourceofmagnonswasrespon- γ∗ = J/~N ∼ 1012/N Hz. This gives the optimal value sible for changing the magnon current and the magnon ofγ belowwhichthefluxshouldbeballistic. Lettingh= density. Consequentially, the transition between the on µ B, where µ is the Bohr magnetron, we find that the B B and off states was in their case quite smooth. Here we critical magnetic field B∗ where the plateau transition seeanextremelyabrupttransition,thusbeingpotentially occurs is, in correct units, |B∗|(T) ≃ J/µ N ≃ 10/N. B more suited for a logic gate. Hence, for any reasonable values of N, very small magnetic fields may suffice to induce the plateau transition. In what concerns an experimental realization of the In summary we have studied the quantum Heisenberg presentidea,itisimportanttonotethateventhoughwe chain driven by two Lindblad baths and subject to two studied a very specific situation, the underlying physi- magnetic fields acting on each boundary. An exact solu- calprinciples ofourresults areverygeneral,being based tion was given in terms of matrix product states which only on the entrapment of magnons by magnetic fields. enables one to calculate local observables for any chain Hence, similar results should be obtained in different size. Thesystemisseentoundergoadiscontinuoustran- field configurations which maintain the same principles. sition from ballistic to sub-diffusive spin current as a Most magnonic circuits are constructed using Yttrium function of the field intensity. Thus, the system may iron garnet (YIG),19,35 which is well described by the function as an extremely sensitive magnonic logic gate Heisenbergmodel,albeitwithadifferentspinvalue. The using the boundary fields as the base. ACKNOWLEDGMENTS Lindblad generators then represent microstrip antennas which are used to generate and collect magnons.19,20 Even though the Lindblad dissipators have been exten- The authors would like to acknowledge the Saõ Paulo sively used in the past to study open quantum systems, Funding Agency (FAPESP) and SPIDER for the finan- we are unaware of any papers mentioning this specific cial support. ∗ [email protected] Physical Review E 87, 012109 (2013). 1 D. J. Evans, E. G. D. Cohen, and G. P. Morriss, 10 G. T. Landi and M. J. de Oliveira, Physical ReviewLetters 71, 2401 (1993); D.J.Evansand Physical Review E 87, 052126 (2013); D. J. Searles, Physical ReviewE 50, 6 (1994). Physical Review E 89, 022105 (2014). 2 G. Gallavotti and E. G. D. Cohen, 11 D. Karevski and T. Platini, Physical ReviewLetters 74, 2694 (1995); Physical Review Letters 102, 207207 (2009); Journal of Statistical Physics 80, 931 (1995). T. Platini, R. J. Harris, and D. Karevski, 3 C. Jarzynski, Physical Review Letters 78, 2690 (1997); Journal of Physics A: Mathematical and Theoretical 43, 135003 (2010) Physical ReviewE 56, 5018 (1997); 12 M. Terraneo, M. Peyrard, and G. Casati, Journal of Statistical Physics 98, 77 (2000). Physical Review Letters 88, 094302 (2002); 4 G.E.Crooks,Journal of Statistical Physics 90, 1481 (1998); B. Li, L. Wang, and G. Casati, Physical ReviewE 61, 2361 (2000). Physical Review Letters 93, 184301 (2004). 5 P. Talkner, E. Lutz, and P. Hänggi, 13 C. W. Chang, D. Okawa, A. Majumdar, and A. Zettl, Physical ReviewE 75, 050102 (2007); Science 314, 1121 (2006). P. Talkner, M. Campisi, and P. Hänggi, 14 G.T.Landi,E.Novais,M.J.deOliveira, andD.Karevski, Journal of Statistical Mechanics: Theory and Experiment 2009, PP0h2y0s2ic5al(2R0e0v9i)e;w E 90, 042142 (2014). M. Campisi, P. Hänggi, and P. Talkner, 15 S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Reviews of Modern Physics 83, 771 (2011). Daughton,S.vonMolna´r, M.L.Roukes,A.Y.Chtchelka- 6 P. Drude,Annalen der Physik 306, 566 (1900). nova, and D. M. Treger, Science 294, 1488 (2001). 7 Z. Rieder, J. L. Lebowitz, and E. Lieb, 16 S. Murakami, N. Nagaosa, and S.-C. Zhang, Journal of Mathematical Physics 8, 1073 (1967). Science 301, 1348 (2003). 8 M. Bolsterli, M. Rich, and W. M. Visscher, 17 I. Zˇutić, J. Fabian, and S. Sarma, Physical ReviewA 1, 1086 (1970). Reviews of modern physics 76, 323 (2004). 9 D. Manzano, M. Tiersch, A. Asadian, and H. J. 18 M. Oltscher, M. Ciorga, M. Utz, Briegel, Physical Review E86, 061118 (2012); A. Asa- D. Schuh, D. Bougeard, and D. Weiss, dian, D. Manzano, M. Tiersch, and H. J. Briegel, Physical Review Letters 113, 236602 (2014). 7 19 A. A. Serga, A. V. Chumak, and B. Hillebrands, 29 V. Popkov and R. Livi, Journal of Physics D: Applied Physics 43, 264002 (2010). New Journal of Physics 15, 023030 (2013). 20 A. V. Chumak, A. A. Serga, and B. Hillebrands, 30 Y. Yan, C.-Q. Wu, and B. Li, Nature communications 5, 4700 (2014). Physical Review B 79, 014207 (2009); L. Zhang, 21 M. Bandyopadhyay and D. Segal, Y. Yan, C.-Q. Wu, J.-S. Wang, and B. Li, Physical ReviewE 84, 011151 (2011). Physical Review B 80, 172301 (2009). 22 G. Benenti, G. Casati, T. Prosen, D. Rossini, and 31 G.Lindblad,Communications in Mathematical Physics 48, 119 (1976) M. Zˇnidariˇc, Physical Review B 80, 035110 (2009). 32 H.-P. Breuer and F. Petruccione, The Theory of Open 23 M. Zˇnidariˇc,Physical Review Letters 106, 220601 (2011); Quantum Systems (Oxford University Press, USA, 2007) Physical ReviewB 88, 205135 (2013). p. 636. 24 T. Prosen and M. Zˇnidariˇc, 33 The Heisenberg Hamiltonian prohibits 3 magnon events Physical ReviewB 86, 125118 (2012). since it conserves the number of excitations.36 Events in- 25 J. J. Mendoza-Arenas, S. Al-Assam, volvingmorethan4magnonsareinprinciplepossible,but S. R. Clark, and D. Jaksch, much less likely. In real systems 3-magnon eventsdo exist Journal of Statistical Mechanics: Theory and Experiment 2013, Pas07a0c0o7n(s2e0q1u3e)n;ce of more complex interactions (e.g. dipo- J. J. Mendoza-Arenas, T. Grujic, D. Jaksch, and S. R. lar). However, as shown in Ref. 20, 4-magnon events are Clark, Physical Review B 87, 235130 (2013). themost relevant type,at least for YIG. 26 T. Prosen, Physical Review Letters 107, 137201 (2011); 34 Z.LenarcicandT.Prosen,arXiv (2014),arXiv:1501.0239. Physical ReviewLetters 106, 217206 (2011). 35 R.L. Douglass, Physical Review 120, 1612 (1960). 27 D. Karevski, V. Popkov, and G. Schu¨tz, 36 A. G. Gurevich and G. A. Melkov, Physical ReviewLetters 110, 047201 (2013). “Magnetization oscillations and waves,” (1996). 28 V. Popkov, D. Karevski, and G. Schu¨tz, Physical ReviewE 88, 062118 (2013).