Magnon-polaron formation in XXZ quantum Heisenberg chains
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Magnon-polaron formation in XXZ quantum Heisenberg chains
D. Morais, F. A. B. F. de Moura W. S. Dias
Instituto de F´ısica, Universidade Federal de Alagoas, 57072-900 Macei´o, Alagoas, Brazil
We study the formation of magnon-polaron excitations and the consequences of different timescales between the magnon and lattice dynamics. The spin-spin interactions along the 1D latticeare ruled by a Heisenberg Hamiltonian in the anisotropic form XXZ, in which each spin exhibitsa vibrational degree of freedom around its equilibrium position. By considering a magnetoelasticcoupling as a linear function of the relative displacement between nearest-neighbor spins, resultsprovide an original framework for achieving a hybridized state of magnon-polaron. Such state ischaracterized by high cooperation between the underlying excitations, where the traveling or station-ary formation of magnon-polaron depends on the effective magnetoelastic coupling. A systematicinvestigation reveals the critical amount of the magnon-lattice interaction ( χ c ) necessary to emer-gence of the stationary magnon-polaron quasi-particle. Different characteristic time scales of themagnon and the vibrational dynamics unveiled the threshold between the two regimes, as well asa limiting value of critical magnetoelastic interaction, above which the magnon velocity no longerinterferes at the critical magnetoelastic coupling capable of inducing the stationary regime. PACS numbers:
I. INTRODUCTION
Collective excitations in magnetic ordering of a mate-rial result in the well-known spin waves, whose the quantaare called magnon. Prospects of using magnons for wave-based computation and data transfer have been stud-ied [1–5], where recent advances such as magnon transis-tors [1], spin-Hall oscillators [3] and spin-wave diodes [4]are reported. Although magnonics is a promising ap-proach [6, 7], designing and controlling such quantumprocesses for long-time dynamics primarily requires un-derstanding the role played by different ingredients.Lattice vibrations are a key condensed matter topic,which have been remarkable effect to charge transport inpolymers [8] and molecular crystals [9, 10]. Although thefirst studies of magnetoelastic coupling of magnons andlattice vibrations have been developed some time ago [11–13], such aspect has been actively studied on both ex-perimental [14–23] and theoretical framework [24–27].Studies of magnetoelastic coupling in ferromagnetic man-ganese perovskites have shown that the spin-wave soft-ening and broadening are related to nominal intersec-tion of the magnon and optical phonon modes [14]. In-elastic neutron scattering was used to study low-energyferromagnetic magnons and acoustic phonons in the fer-rimagnetic insulators, whose results provided evidencefor the presence of magnon-phonon [16]. Signaturesof interaction between spin and phonon have been re-ported for magneto-thermal transport measurements inp-doped Si [17], as well as layered semiconducting fer-romagnetic compound CrSiTe explored by raman scat-tering experiments [18]. Probing the interaction betweenmagnon and lattice vibrations revealed a conversion ofspin wave-packets into an elastic wave-packets in filmsof a ferrimagnetic insulator under non-uniform magneticfields [19]. The magnon-phonon coupling and the forma-tion of an optically excited magnon-polaron with high co-operation were described for a metallic ferromagnet with a nanoscale periodic surface pattern, where symmetriesof the localized magnon and phonon states have been re-ported decisive for the hybridized state formation [20]. Astrong coupling between magnons and phonons has beendetected in the thermal conductivity of antiferromagnet Cu T eO [22]. Magnetoelastic coupling has been theo-retically studied to excite spin waves in magneto strictivefilms through surface acoustic waves on piezoelectric sub-strates, in which driven spin waves were able to propagateup to 1200 µm [24]. A theoretical study shows the pos-sibility of generating magnon-phonon coupling througha variable magnetic field in space, whose interaction de-pends on the strength of the magnetic field gradient [26].The problem of a two-dimensional antiferromagnet at thepresence of magnetoelastic coupling it was investigated inref. [27] . The authors demonstrate that the magnon andphonon bands are hybridized due to the magnon-phononcoupling, with the properties of the magnon-phonon ex-citations suggesting a nontrivial SU (3) topology.Another exciting experimental development involvinginteraction between magnons and lattice vibrations werethe spin Seebeck effect [28–30] and the bottleneck ac-cumulation of hybrid magnetoelastic bosons [31]. Thefirst one describes a spin current that appear in mag-netic metal systems under effect of a thermal gradient,whose effect is enormously enhanced by non-equilibriumphonons [30]. The bottleneck accumulation phenomenonfor a magnon-phonon gas demonstrates how magnon-phonon scattering can significantly modify a formation ofa Bose-Einstein condensate of an ensemble of magnons,providing a novel condensation phenomenon with a spon-taneous accumulation of hybrid magnetoelastic bosonicquasiparticles [31].Previous studies exemplifies how the interaction be-tween magnons and vibrational lattice modes turned intoa hot topic of research, considered as a powerful methodeither for spin control, as well as potential use as trans-duction from magnon signals to electrons [32]. Still onthe promising character of the magnonics, we observethe recent advent of advanced materials exhibiting high-frequency magnons, which have been impelled the devel-opment of a new class of ultrafast spintronic devices [33–36]. Such studies have reported terahertz magnons inthe 2D Ising honeycomb ferromagnet CrI [33], ultra-thin film of iron-palladium alloys [34], layered iron-cobaltmagnonic crystals [35] and noncollinear magnetic bilay-ers [36]. As we consider all the previously aspects, weare faced with the question: How does magnetic exci-tation behave under different time scales of the magnonand the lattice vibrations? In fact, the magnon exci-tation in a magnetoelastic lattice has an interdependentrelaxation mechanism, and the formation of the magnon-polaron lacks a greater understanding. What would bethe consequences of a spin transport as fast as the lat-tice dynamics? In order to answer these questions, weoffer a systematic investigation of a quantum Heisen-berg model, in which each spin of a one-dimensional lat-tice exhibits a vibrational degree of freedom around itsequilibrium position. Such character is described by astandard Hamiltonian of coupled harmonic oscillators,whose magnetoelastic coupling is described by an ex-change interaction which depends on the lattice defor-mations. By considering an intrinsic anisotropy medi-ated magnetoelastic coupling, we explore the Heisenbergspin-spin coupling within a XXZ framework, whose re-sults demonstrate an original method for achieving a hy-bridized state referred as magnon-polaron. Such state ischaracterized by high cooperativity between the under-lying excitations, in which a traveling or stationary for-mation depends on the magnetoelastic interaction. Wereveal the critical amount of the magnon-lattice interac-tion ( χ c ) necessary to emergence of the static magnon-polaron quasi-particle. Bellow the critical magnetoelas-tic interaction, the magnon-polaron excitation developstwo-fronts that propagates with constant velocity, whosespatial matching of their wave distributions exhibits aselection of particular modes. Their velocities continu-ously decreases with the power-law dependence as themagnon-lattice interaction grows. By exploring the ratiobetween the characteristic time scales of the magnon andthe vibrational lattice, we unveil a limiting value of thecritical magnetoelastic coupling, which is achieved as themagnon dynamics becomes much slower than the latticedynamics. II. MODEL AND FORMALISM
The problem consist in analyzing a one-dimensionalmagnetoelastic lattice, in which spins 1 / H = H mag + H latt , (1) FIG. 1: Pedagogical representation of the unidimensionalspin lattice with effective harmonic springs coupling nearest-neighbor spins. The exchange interaction along the field direc-tion ( J Zn, +1 ) depends linearly on the distance between spins.Thus, the magnetic excitation contributes to the emergence ofvibrational components, whose simultaneity and interactionmediated by the magnetoelastic coupling excites the magnon-polaron formation. with the vibrational contribution of system H latt givenby H latt = X n p n M + κ u n +1 − u n ) . (2)Here, M represents the mass of the ions and κ is the effec-tive spring’s constant. Further, p n = M ˙ u n describes theconjugated momentum for the n -th spin. By consider-ing the spin lattice along the x axis, we parameterize theHamiltonian in terms of the displacement u n = a ′ n − a n ,with a ′ n and a n denoting the respective position and equi-librium position of ion n . We consider ω ¯ h << k B T , inwhich ω = p κ/M , k B is the Boltzmann constant and T the temperature. In this framework, the lattice dynamicscan be treated within the classical mechanics formalism.The magnetic component is governed by a quantumHeisenberg Hamiltonian, in which the spin-spin interac-tions are described by a nearest-neighbor exchange. Westudy the XXZ model H mag = E + gµ B N H + X n hS ( J Zn,n +1 + J Zn,n − ) c † n c n − hSJ XYn,n +1 c † n +1 c n − hSJ XYn,n − c † n − c n , (3)with S = ¯ h/ J Zn,m denoting the exchange anisotropiccomponent between the n -th and m -th spins, and J XYn,m describing the exchange coupling components in theXY plane. The ground-state energy in the presenceof a uniform external magnetic field is given by E = − gµ B N HS − S P n J Zn,n ± . Here, we will focus on thepropagation of magnetic excitation. Thus, c † n and c n arerespectively the creation and annihilation operators atthe n -th site. Whenever the creation operator is appliedto the ground-state, it leads to the excited state with thespin at site n flipped. Furthermore, we consider an intrin-sic anisotropy-mediated magnetoelastic coupling, suchthat the spin-spin coupling at the “Z” direction (i.e thedirection of external magnetic field) depends on the ef-fective displacements between neighboring spins. We as-sume a regime of small amplitude oscillations describedby J Zn,n +1 = J + α ( u n +1 − u n ) J XYn,n +1 = J , (4)where α denotes the effective spin-lattice coupling affect-ing the longitudinal spin-spin interactions. Such frame-work gives a ground-state energy ( E = − gµ B N HS − S N J ) independent from the vibrational modes of thespin chain.We analyze two key parameters: the effective cou-pling between the magnetic properties and the vibra-tional modes (i.e. the magnon-lattice interaction) χ =¯ h α /J κ ; and the ratio of the characteristic times scalesof magnon ( t m = 1 / ¯ hJ ) and ionic chain ( t l = 1 /ω ), writ-ten as τ = t m /t l . By employing a normalized spin posi-tion x n = q κ/ ¯ h J u n , the set of equations that describethe dynamics of the magnon and the lattice vibrationscan be written respectively as it m ˙ ψ n = 2 ψ n − ψ n +1 − ψ n − − √ χ ( x n +1 − x n − ) ψ n and t m τ ¨ x n = x n +1 + x n − − x n − √ χ ( | ψ n − | − | ψ n +1 | ) . (5)The above set differential equations was solved by us-ing a standard Runge-Kutta method with a time stepsmall enough to keep the wave function norm conser-vation ( | − P n | ψ n | | ≤ − ) along the entire timeinterval considered. We concentrate our study by con-sidering the initial state as single spin flip fully localizedand centered at rest in static lattice center ( n = 0 will betaken as the center of chain). Furthermore, we considerthe characteristic time scale t m as the relevant time unit.We explore the regime with τ > III. RESULTS
Using the numerical method described above, we startby examining the time evolution of initially fully localized
FIG. 2: Time evolution of the magnon wave-function | ψ n | versus time and n ( n = 0 represents the center of chain).The local deformation x n +1 − x n − of chain is also investi-gated. Calculations were done considering τ = 10 . . (a) Inthe absence of magnon-lattice coupling ( χ = 0 .
0) the chainremains static and the magnon propagates ballisticaly alongthe chain (b-c) considering χ = 0 . χ = 1 . magnon wave-packet at the center of a lattice initially inrest ( x n = 0 and ˙ x n = 0), with τ = 10 . . In Figure 2we plot the time evolution of the wave-function profile | ψ n | and its respective lattice deformation x n − x n − forsome representative values of magnetoelastic coupling χ .In the absence of magnon-lattice interaction ( χ = 0),we observe the magnon wave-function spreading ballis-tically over the entire lattice, that remains static. Thescenario is significantly modified when we consider the in-teraction between magnon and lattice. For weak magne-toelastic coupling, breathing magnon modes emerge andpropagate with constant velocity, while a fraction of themagnon wave radiates through the lattice (see Figs. 2b-c). By following its respective lattice deformation, we ob-serve signatures of the magnon-polaron formation, whichexplains the non-dispersive profile of the magnon wave-function. Magnon-polaron modes become slower as weincrease the magnetoelastic coupling. A strong enoughcoupling induces a significant fraction of the magneticexcitation to remains trapped around its initial location(see Fig. 2d). Such behavior is also characterized by aspatial matching between the spin-mode and lattice de-formations distribution. Thus, we observe a high cooper-ation between the underlying excitations as a signatureof the hybridized magnon-polaron state.In order to better understand this rich set of dynami-cal profiles, we explore the participation function for the χ 〈 ξ ( t → ∞ ) 〉 τ = 10 τ = 10 τ = 10 χ 〈 Ξ ( t → ∞ ) 〉 τ = 10 τ = 10 τ = 10 ( a )( b ) FIG. 3: Average participation function for both magnon andlattice vibrations at the long time limit versus χ , exploring τ = 10 . up to 10 . . Data suggest a kink singularity de-veloped at χ c , above which the magnon-polaron formationstays stationary. Furthermore, such critical magnetoelasticcoupling χ c changes with different τ . magnon ξ ( t ) = X n / | ψ n ( t ) | , (6)and the lattice vibrationsΞ = (cid:16)P n ( x n +1 − x n ) + ˙ x n τ (cid:17) P n h ( x n +1 − x n ) + ( x n − x n − ) + ˙ x n τ i . (7)Such traditional quantities provide, respectively, an esti-mate of the number of lattice sites over which the magnonwave-function is spreading at time t , and the number ofdisturbed lattice sites at time t . Their scaling behav-ior can be used to distinguish the different dynamicalregimes. The asymptotic participation function becomessize-independent for localized wave packets. On the otherhand, h ξ ( t → ∞ ) i ∝ N and h Ξ( t → ∞ ) i ∝ N corre-sponds to the regime where the magnon wave-packet andthe lattice vibrations are uniformly distributed over thelattice. In fig. 3 we compute the long-time behavior ofthe participation function for the magnon [ h ξ ( t → ∞ ) i ] | ψ n | t = 1000 t m t = 1500 t m t = 2000 t m -7000 -3500 0 3500 7000 n -0.1500.15 x n + − x n − τ v l a tt ( f ) τ 〈 P i n i ( t → ∞ ) 〉 P = Ξ P = ξ ( a ) ( c )( b ) ∼τ FIG. 4: With χ = 1 . τ = 10 . , the spatial profile ofthe magnon wave-function and lattice deformations describethe spatial matching and bound dynamics of the magnon-polaron formation, while (b) the wave-front velocity of thelattice vibrations versus τ shows a linear increasing with τ .(c) The asymptotic participation functions around the initialmagnetic excitation reveals a more pronounced spreading as τ increases. Thus, we observe as the lattice vibrations con-tribute to the magnon trapping and the consequent formationof stationary magnon-polaron. and the lattice dynamics [ h Ξ( t → ∞ ) i ] versus the ef-fective magnetoelastic coupling χ . Here, we explore thetime scales of magnon and lattice vibrations. Calcula-tions were done for τ = 10 . up to 10 . . Magnon andlattice vibrations decrease the propagation as the mag-netoelastic coupling increases. This aspect corroboratesthe previous scenario of magnon-polaron formation, char-acterized by non-dispersive modes of spin and lattice vi-brations exhibiting a constant velocity that decays as themagnetoelastic coupling increases. We further note anemergence of a kink singularity, which reveals an abruptdecreasing at participation functions as χ increases evenmore. Such behavior signals the critical point that es-tablishes the beginning of the stationary regime, corrobo-rated by full agreement exhibited between the asymptoticdynamics of the magnon and the lattice vibrations. Fur-thermore, the long-time participation function is vanish-ingly small as the magnetoelastic coupling increases evenmore, i.e. the degree of trapping is enhanced. The crit-ical magnetoelastic coupling increases as the τ = t m /t L grows, a consequence of the propagation of vibrationalmodes.In Fig. 4a we display snapshots of the magnon wave-packet and the corresponding lattice deformations for χ = 1 . τ = 10 . . Besides magnon and latticedeformations exhibiting non-dispersive modes with per- χ 〈 R ( t → ∞ ) 〉 χ 〈 ρ ( t → ∞ ) 〉 τ = 10 τ = 10 τ = 10 ( a )( b ) ∼(χ−χ c ) FIG. 5: The asymptotic return probabilities of the (a)magnon and (b) the lattice vibrations exhibit clear signaturesof a phase transition between the traveling and stationaryregime of magnon-polaron formation, which corroborates theresults at the Fig. 3. Above the critical magnetoelastic cou-pling χ c , magnon and lattice vibrations become significantlyclustered around the site of the initial magnetic excitation. fect spatial match, the lattice deformations are spread-ing over the lattice by developing wave-fronts. Withthe magnon approximately as fast as the lattice dy-namics, disturbances originating from the lattice wave-fronts that extend along the tails inhibit the propaga-tion of the magnon wave-packet. Thus, a smaller cou-pling between magnetic and mechanical components isrequired to the stationary formation of the magnon-polaron. This character is better understood when look-ing at Fig. 4b-c, where we remain with χ = 1 .
0, butwe explore a range of τ . Fig. 4b shows the wave-frontvelocity of the lattice deformations exhibiting a lineargrowth with τ . Fig. 4c displays the asymptotic participa-tion functions around the initial site of magnon excitation[ n − ≤ n ≤ n + 100]. A small ratio between charac-teristic time scales of magnon and lattice vibration favorsthe stationary magnon-polaron formation, described by h P ini ( t → ∞ ) i ≈
1. With the magnon spreading slowlyenough, the magnetoelastic coupling is unable to estab-lish stationary formation.We also explore the asymptotic regime of the return χ 〈 v t r a ( t → ∞ ) 〉 τ = 10 τ = 10 τ = 10 -3 -2 -1 χ c −χ -1
850 875 900 925 n no r m a li ze d a m p lit ud e sech [λ( n − v tra t / t m )]| x n +1 − x n −1 ||ψ n | -20 -10 0 10 20 n ( a )( b ) ∼(χ c −χ) ( c ) χ = 1.5χ = 1.0 FIG. 6: Dynamics of the traveling formation of magnon-polaron, with (a) the velocity versus χ and its respective scal-ing analysis. The best power-law fit provides v ∝ ( χ c − χ ) / .The spatial profile of the magnon wave-function and thematching lattice deformation for the magnon-polaron state,whether (b) traveling or (c) stationary, corroborates the char-acteristic spatial profile from the breathing bright solitons (fit-ting curves in solid lines). probabilities for the magnon and the lattice deformation: R ( t ) = | ψ n =0 ( t ) | and ρ ( t ) = | x ( t ) − x − ( t ) | . (8)Such measures offer the probability of finding the magnonwave-packet or the lattice deformations at the positioncorresponding to the initial magnetic excitation. Thus,their scaling behaviors can also be used to distinguish be-tween localized and delocalized wave packets in the long-time regime, with R ( t → ∞ ) → ρ ( t → ∞ ) → χ < χ c , the magnon and the latticedeformations exhibit a vanishingly small return proba-bility, which confirms a predominant spreading throughthe lattice. Above a critical magnetoelastic coupling,the asymptotic return probabilities become significantlylarger than 1 /N . Allied to the monotonic growth of bothreturn probabilities, such behavior reinforces the emerg-ing self-trapping regime described earlier. Both quan-tities exhibit a trend ( χ − χ c ) . , as well as confirm a τ χ c -2.0 -1.0 . −χ c stationary magnon-polarontraveling magnon-polaron χ c = 1.825 ∼τ −2 FIG. 7: Plot of χ c versus τ phase diagram. Corroborat-ing previous results, the magnetoelastic coupling necessaryto the stationary formation of magnon-polaron increases asthe τ growth. However, such this behavior is restricted tothe small enough τ regime. The further increasing in the ra-tio of characteristic time-scales unveils a limit value χ maxc ,above which the critical coupling becomes indifferent to the τ . The best fitting is achieved for χ maxc = 1 . χ maxc − χ c ) ∝ τ − (see inset). dependence on the parameter τ (see inset).The magnon-polaron formation and its threshold be-tween traveling and stationary regimes have also beenidentified by exploring the magnon-polaron velocity. Forthe Fig. 6a, we measure the mean velocity of the magnon-polaron modes at the long-time regime [ h v tra ( t → ∞ ) i ]and investigate its relationship with χ . Data show thevelocity decreasing as the effective magnetoelastic cou-pling increases, up to the threshold at which vanishes( χ ≥ χ c ). Besides the critical magnetoelastic couplingshowing full agreement with the measures of participa-tion function (see Fig. 3) and return probability (seeFig. 5), the traveling velocity of the magnon-polaron de-creases with h v tra ( t → ∞ ) i ∝ ( χ c − χ ) . , the same expo-nent exhibited by the asymptotic return probability afterthe critical point.In order to better understand such hybridized exci-tation of magnon-polaron, in Fig. 6b-c we show pro-files of the magnon wave-function and the matching lat-tice deformation in a snapshot achieved during the time-evolution for a system ruled by τ = 10 . and χ = 1 . , . [ λ ( n + vt/t m )]) [9, 10], either for the traveling(see Fig. 6b) and the stationary excitations (see Fig. 6c).Such magnetic components are bound to a lattice struc-tural kink that also exhibits the well-known breathingbright soliton-like spatial profile.The consequences of different time scales of the mag-netic and vibrational components are exhibited in Fig. 7,in which we extend our numerical experiments in or-der to offer χ c versus τ diagram. For greater accu- racy, data have been computed by analyzing the par-ticipation function, the return probability, as well as themagnon-polaron velocity. Systems in which the dynamicsof magnons is comparable to the lattice dynamics showan increase in the critical magnetoelastic coupling χ c as τ increases. However, when considering systems withan ever slower magnon dynamics, this behavior leadsmonotonically χ c to a limit value [ χ c ≈ . t m /t L ratio becomes indiffer-ent. By analyzing the critical magnetoelastic couplingversus τ , the best fitting provides χ maxc ≈ . χ maxc − χ c ) ∝ τ − (see inset). IV. SUMMARY
In this work we study how the lattice dynamics influ-ence the dynamics of initially localized one-magnon exci-tations. We consider a quantum anisotropic Heisenbergferromagnetic chain, in which the spins 1 / V. ACKNOWLEDGMENTS
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