Conservation of Angular Momentum in the Elastic Medium with Spins
CConservation of Angular Momentum in the Elastic Medium with Spins
Dmitry A. Garanin and Eugene M. Chudnovsky
Physics Department, Herbert H. Lehman College and Graduate School, The City University of New York,250 Bedford Park Boulevard West, Bronx, New York 10468-1589, USA (Dated: January 5, 2021)Exact conservation of the angular momentum is worked out for an elastic medium with spins.The intrinsic anharmonicity of the elastic theory is shown to be crucial for conserving the totalmomentum. As a result, any spin-lattice dynamics inevitably involves multiphonon processes andinteraction between phonons. This makes transitions between spin states in a solid fundamentallydifferent from transitions between atomic states in vacuum governed by linear electrodynamics.Consequences for using solid-state spins as qubits are discussed.
The problem of transfer of angular momentum betweenmagnetic moments and macroscopic body goes back toseminal experiments of Einstein – de Haas [1] and Bar-nett [2]. The first established that the change in the mag-netization of a freely suspended body is accompanied bymechanical rotation. The second demonstrated that ro-tation causes magnetization. Macroscopic explanation ofthese phenomena is straightforward – based upon conser-vation of the total angular momentum. Equally straight-forward is microscopic theory of spin-phonon processesdeveloped in seminal papers of Van Vleck [3]. Magne-toelastic Hamiltonians studied by classical and quantummethods for specific materials ever since have been writ-ten to reflect anisotropy of the crystal lattice. Due tothe lack of rotational symmetry they do not conserve thetotal angular momentum. Until recently this inconsis-tency was swept under the rug by correctly assuming thatany unaccounted change in the angular momentum wouldbe absorbed by the whole body. Attempts to demon-strate this in a rigorous manner [4–6], while conceptuallyvaluable, have been mathematically cumbersome with noclear consequences for experiments.In recent years this problem received renewed atten-tion due to the emergence of spintronics and nanoelec-tromechanical devices, as well as due to the prospectof developing spin-based quantum computers. Univer-sal parameter-independent nature of spin-lattice relax-ation arising from the conservation of the total angularmomentum has been demonstrated [7]. Transfer of theangular momentum from spins to mechanical degrees offreedom in nanomechanical oscillators has been studiedtheoretically [8–13] and experimentally [14–17]. The di-vision of the phonon angular momentum into orbital andspin parts was suggested [18] and further explored in ap-plication to problems of spin relaxation [19, 20] and spintransport [21]. The concept of angulon, a quasiparticlecarrying an angular momentum, initially introduced todescribe properties of molecular impurities in a super-fluid [22], has been extended to magnetic impurities insolids [23]. The physics of the Einstein - de Haas ef-fect in magnetic insulators has been recently revisitedwithin a model that decouples rotations from vibrationsand separates variables responsible for microscopic and macroscopic mechanical torques [24].In Ref. 19 we demonstrated conservation of the angu-lar momentum by spin-lattice interaction for a specificquantum problem of a relaxing spin. The general caseturned out to be more subtle even at the classical level.It will be solved here and will elucidate the fact thatspin-phonon processes are fundamentally different fromspin-photon processes due to the intrinsic nonlinearity ofthe elastic theory as compared to electrodynamics.We begin with underappreciated derivation of the con-servation of angular momentum in a conventional elastictheory that is not easy to find in textbooks. The expres-sion for the angular momentum of the elastic solid thatis linear on a small deformation u ( r , t ) is: L (0) = ˆ d r ( r × p ) . (1)Here p ( r , t ) = ρ ˙ u ( r , t ) is the momentum density, with ρ being the mass density of the solid. The time derivativeof this expression yeilds: ˙ L (0) = ˆ d rρ ( r × ¨ u ) . (2)The second time derivative of u in this equation satisfiesthe Newton’s equation, ρ ∂ u α ∂t = ∂σ αβ ∂r β , (3)with the force in the right-hand-side being the gradient ofthe stress tensor σ αβ = δ H /δe αβ . Here H is the Hamil-tonian of the system and e αβ = ∂u α /∂r β is the straintensor. Substituting Eq. (3) into Eq. (2) and integratingby parts under the assumption of zero elastic stress atthe free boundary of the body, we obtain ˙ L (0) α = ˆ d r(cid:15) αβγ r β ∂σ γδ ∂r δ = − ˆ d r(cid:15) αβγ σ γβ , (4)where summation over repeating indices is assumed with (cid:15) αβγ being an absolutely antisymmetric unit tensor ofthird rank (Levi-Civita symbol). In the linear theory ofelasticity that ignores local internal torques associated a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n with spins the Hamiltonian is H = ˆ d r (cid:18) ρ ˙ u + 12 C αβγδ u αβ u γδ (cid:19) (5)with u αβ = ( e αβ + e βα ) and tensor C αβγδ reflecting thesymmetry of the crystal lattice. This makes the stresstensor σ γβ in Eq. (4) proportional to the strain (Hooke’slaw) and symmetric with respect to the transposition of γ and β . The product of the antisymmetric tensor (cid:15) αβγ and the symmetric tensor σ γβ in the last of Eq. (4) isautomatically zero, which makes ˙ L (0) zero and provesconservation of the angular momentum in the limit linearon deformations.This however is not the end of the story if one recallsthe exact definition of the symmetrized strain tensor, u αβ = 12 ( e αβ + e βα + e γα e γβ ) , (6)that follows from the analysis [25] of how deforma-tions change distances between two points in the elasticmedium: dl (cid:48) = dl + 2 u αβ dr α dr β . (7)With that definition of u αβ one obtains σ αβ = δ H δe αβ = δ H δu αβ + e αγ δ H δu γβ . (8)The first tensor in this expression is symmetric but thesecond is not, even for an isotropic body, which makes ˙ L in Eq. (4) nonzero. Solution to this problem comes fromthe realization that the full expression for the angularmomentum in the elastic medium is L = ˆ d r ( r + u ) × p = ˆ d rρ ( r + u ) × ˙ u . (9)The term quadratic on u , that Zhang and Niu [18] associ-ated with the phonon spin, comes from the same necessityto distinguish between coordinates of atoms before andafter deformation that leads to Eq. (6) via Eq. (7). Thetime derivative of the angular momentum now becomes ˙ L α = − ˆ d r ( (cid:15) αβγ σ γβ + (cid:15) αβγ e βδ σ γδ ) . (10)Substituting Eq. (8) into Eq. (10) and working out tensorproducts that contain zero convolutions of symmetric andantisymmetric tensors, one obtains ˙ L = 0 , which provesthe exact conservation of the angular momentum in theconventional nonlinear elastic theory to all orders on thedeformation.We shall now include into the theory a single atomicspin located at a point r = r . This immediately takesus outside the framework of the conventional elasticitybecause spins, as they rotate, generate internal torques ! Figure 1: Schematic illustration of the elastic twist due to thereaction of the atomic lattice to a rotating spin. (see Fig. 1) that are explicitly neglected by the elastictheory. As we shall see, however, this problem can befixed in the same manner as the linear theory of elasticitywas fixed before.Consider for certainty a uniaxial spin Hamiltonian H S = H S δ ( r − r ) , H S = F ( n · S ) , (11)where F is an arbitrary scalar function of its argu-ment, e.g., F = − D ( n · S ) , with n being the mag-netic anisotropy axis in the non-deformed state. Elas-tic deformations of the body change the direction ofthe anisotropy axis n . Local rotation by a small angle δ φ = ∇ × u changes n to n (cid:48) = n + δ φ × n + 12 δ φ × ( δ φ × n ) + ... (12)In the first order on deformation it results in the spin-lattice interaction of the form δH S = ∂F∂ n · ( δ φ × n ) . (13)It is difficult, however, to develop rigorous approach inall orders on δ φ needed to prove exact conservation ofthe angular momentum.Below we adopt a different approach by noticing thatthe unit vector n determined by the symmetry of thecrystal lattice goes in the direction of the vector connect-ing two points in a solid. Deformation changes it as n = d r dl ⇒ n (cid:48) = d r (cid:48) dl (cid:48) , (14)where dr (cid:48) α = dr α + du α = dr α + e αβ dr β (15)and dl = | d r | and dl (cid:48) = | d r (cid:48) | are the lengths of the in-finitesimal vector d r before and after deformation, relatedthrough Eq. (7). From these formulas one obtains n (cid:48) α = dr (cid:48) α dl (cid:48) = n α + e αβ n β (cid:112) u δη n δ n η . (16)With the replacement of n by n (cid:48) the Hamiltonian in Eq.(11) becomes the exact spin-lattice Hamiltonian account-ing for all orders on the deformation, H S = F ( n (cid:48) · S ) , (17)where n (cid:48) is given by Eq. (16).The spin contribution to the mechanical stress tensoris σ ( S ) αβ = δ H S δe αβ = δ H S δn (cid:48) γ ∂n (cid:48) γ ∂e αβ . (18)After a straightforward algebra we get with the help ofEq. (6) ∂n (cid:48) α ∂e βγ = ( δ αβ − n (cid:48) α n (cid:48) β ) n γ (cid:112) u δη n δ n η . (19)Substituting this into Eq. (18) one obtains from Eq. (10)the spin contribution to the time derivative of the me-chanical angular momentum: ˙ L ( S ) α = − (cid:15) αβγ ( δ βδ + e βδ ) ∂ ˆ H S ∂n (cid:48) η ∂n (cid:48) η ∂e γδ . (20)With the help of Eq. (11), it can be re-written as ˙ L ( S ) α = − F (cid:48) ( n (cid:48) · S ) S η (cid:15) αβγ ( δ βδ + e βδ ) ∂n (cid:48) η ∂e γδ , (21)where F (cid:48) ( x ) = dF/dx . Eqs. (16) and (19) allow one towrite ( δ βδ + e βδ ) ∂n (cid:48) η ∂e γδ = n (cid:48) β (cid:0) δ γη − n (cid:48) η n (cid:48) γ (cid:1) . (22)Dropping the zero convolution of symmetric and anti-symmetric tensors we obtain from Eq (21) ˙ L ( S ) α = − F (cid:48) ( n (cid:48) · S ) S η (cid:15) αβγ n (cid:48) β (cid:0) δ γη − n (cid:48) η n (cid:48) γ (cid:1) = − F (cid:48) ( n (cid:48) · S ) (cid:15) αβγ n (cid:48) β S γ . (23)The equation of motion for the spin is (cid:126) ˙S = − (cid:20) S × δ H S δ S (cid:21) , (24)or, in the tensor form (cid:126) ˙ S α = − (cid:15) αβγ S β δ H S δS γ . (25) Using Eq. (11), one can write it as (cid:126) ˙ S α = F (cid:48) ( n (cid:48) · S ) (cid:15) αβγ n (cid:48) β S γ . (26)Comparing Eqs. (23) and (26) we immediately seethat the time derivative of the total angular momentum, J = (cid:126) S + L ( S ) , is zero. This concludes derivation ofthe exact conservation of the total angular momentum,spins + lattice, in the full nonlinear elastic theory withembedded spins. In that derivation we did not use theexplicit form of the function F ( n · S ) representing thespin Hamiltonian. The same derivation will apply to anyform of the spin-lattice interaction.One observation that follows from our derivation isthat conservation of the total angular momentum in spin-lattice dynamics can only be recovered after one accountsfor all nonlinear terms in the elastic and magnetoelasticparts of the Hamiltonian. This must have consequencesfor quantum theory of spin-lattice interactions as well. Inquantum mechanics deformations are quantized accord-ing to ˆ u ( r ) = (cid:115) (cid:126) ρV (cid:88) k λ e k λ √ ω k λ (cid:16) e i k · r a k λ + e − i k · r a † k λ (cid:17) , (27)where ω k λ and e k λ are frequencies and polarizations ofphonons, V is the volume of the body, and a † k λ , a k λ arephonon operators of creation and annihilation.In the linear elastic theory the strain tensor u αβ = ( e αβ + e βα ) is linear on these operators. The elasticHamiltonian (5) is quadratic on phonon operators whilethe spin-lattice Hamiltonian in the rotational approxi-mation of Eq. (13) is linear on phonon operators. Conse-quently, at low temperature, when thermal phonons areabsent, the spin-lattice theory that describes absorptionand emission of phonons is similar to the theory of atomictransitions in electrodynamics.This changes when one accounts for nonlinear (anhar-monic) terms needed to conserve the angular momen-tum. The exact strain tensor (6) contains terms thatare quadratic on deformation and thus quadratic on thephonon operators. This contributes terms up to thefourth order on phonon operators to the elastic Hamil-tonian (5). As to the exact spin-lattice Hamiltonian, ac-cording to Eqs. (17) and (16) it contains all orders of thedeformation and, thus, all orders of the phonon opera-tors.Consequently, multiphonon processes and interactionbetween phonons, required by the conservation of thetotal angular momentum, inevitably enter the quantumproblem. The intrinsic nonlinearity of the elastic theorymakes the dynamics and manipulation of an atomic spinin a solid fundamentally different from the dynamics andmanipulation of the atomic states in vacuum. The latterare described by the linear electrodynamic theory thatdoes not require multiphoton processes for the conserva-tion of the angular momentum.This does not mean that a revision of the theory ofspin-phonon transition rates is needed at low tempera-ture. When thermal phonons are absent, contribution ofmultiphonon processes to the rates would be proportionalto higher powers of the spin-lattice interaction and, there-fore, would be small. 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