Hamiltonian reduction for the magnetic dynamics in antiferromagnetic crystals
aa r X i v : . [ m a t h - ph ] O c t Hamiltonian redu tion for the magneti dynami s in antiferromagneti rystalsD.O.Sinitsyn(Dated: De ember 15, 2018)The nonlinear spin dynami s in antiferromagneti rystals is studied for the magneti stru turessimilar to that of hematite. For the ase when only two magnetization ve tors are non-zero and theHamiltonian has an axial symmetry, a redu tion to a Hamiltonian system with one degree of freedomis performed, based on the orresponding onservation law. The analysis of the phase portraits ofthis system provides tra table analyti al and geometri des riptions of the regimes of nonlinear spindynami s in the rystal.Keywords: spin, nonlinear dynami s, symmetry, redu tionI. INTRODUCTIONThe phenomenologi al method of des ribing the properties of rystals with magneti order is based on attributing lassi al magnetization ve tors ~S i ( i = 1 , . . . n ) to the magneti sublatti es of a rystal ( n is the number of thesublatti es), [2℄. These ve tors are subje t to ex hange intera tion between themselves, anisotropi intera tion withthe rystal latti e, and, optionally, Zeeman intera tion with external magneti (cid:28)elds, [1℄. The dynami s of thesemagnetizations an be des ribed via the orresponding Hamiltonian system. Generally, the latter is nonlinear and itssolving presents substantial di(cid:30) ulties. In the regimes when the spin ve tors are lose to their equilibrium positions,the linear approximation an be used, whi h allows to obtain the spin wave spe tra in a relatively straightforwardmanner. However, the general nonlinear dynami s of sublatti e magnetizations is also of onsiderable interest, e.g.for nonlinear regimes of magneti resonan e. Hen e there is a need for methods of qualitative and quantitativeinvestigating those dynami s in various situations.To this end, the analyti al tools devised for the study of me hani al Hamiltonian systems an be employed. In thepresent paper we apply an approa h based on the redu tion of a Hamiltonian system to another Hamiltonian systemof lower dimensionality. The key idea is the onsidering of a subalgebra of the Poisson algebra of dynami variables. Ifa subalgebra ontains the Hamiltonian or, more generally, the Hamiltonian depends on the elements of the subalgebraand several (cid:28)rst integrals, then the elements of the subalgebra are subje t to a new Hamiltonian system with a lowernumber of phase variables. It was Poin are who (cid:28)rst performed a redu tion of a similar type in the three-bodyproblem (see [4℄). A redu tion based on onsidering a Poisson subalgebra was applied to the magnetization dynami sin super(cid:29)uid He B in [6℄, lowering the phase spa e dimensionality from 6 to 3.In the present work we onsider the lass of Hamiltonians having an axial symmetry. As a main example we usethe ase des ribed in the lassi al work by Dzyaloshinsky, [3℄, (cid:21) the Hamiltonian omprising the leading terms inthe magneti energy of the four sublatti es of the antiferromagneti rystal α - F e O . It belongs to rhombohedralsystem, having a third-order symmetry axis, [3℄. The magneti energy omprises ex hange terms and several spin-latti e intera tion terms, some of them orresponding to the so- alled Dzyaloshinsky-Moriya (cid:28)eld leading to weekferromagnetism of α - F e O (other possibilities in lude weak additional antiferromagnetism (cid:21) the ase of Cr O in[3℄, and the distortion of magneti stru ture des ribed in [7℄). In our ase, if only se ond-order terms and the largestfourth-order term are taken into a ount (see [3℄, se tion "Ferromagnetism of α - F e O "), then the Hamiltonian isinvariant with respe t to the rotation about the axis of the rystal. This results into the sum of the longitude anglesof the spins being a y li variable, i.e. not entering the Hamiltonian (for a proper hoi e of dynami variables). Thisallows to perform a redu tion to a Poisson subalgebra as outlined above, de reasing the number of phase variables by2 units. In the ase of α - F e O , where only the ferromagneti ve tor and one antiferromagneti ve tor are non-zero,this redu tion leads to a system with one degree of freedom, whi h an be e(cid:27)e tively investigated by means of phaseportraits. That provides a detailed pi ture of the nonlinear regimes of the spin dynami s in the given approximation.The same results hold for the arbonates of Fe, Mn and Co with a di(cid:27)eren e only in the values of the parameters ofthe Hamiltonian.II. THE HAMILTONIAN FORMULATIONWe follow the well-known paper [3℄ by Dzyaloshinsky for onstru ting the Hamiltonian system des ribing the spindynami s in α - F e O . The unit ell of the rystal is rhombohedral. The rystal has 4 magneti sublatti es, and thefour orresponding Fe ions in the unit ell lie on the body diagonal of the rhombohedron (cid:21) the third order symmetryaxis of the rystal. The magnetizations of the sublatti es are denoted ~s , ~s , ~s , ~s . The Poisson bra kets betweentheir omponents have the usual form of the bra kets for angular momentum: { s iα , s iβ } = ε αβγ s iγ , { s iα , s jβ } = 0 , i = j, (1)where the Greek letters denote the Cartesian oordinates of the ve tors (here and below repeated indi es implysummation over 1,2,3). The degenera y of these bra kets leads to the existen e of four Casimir fun tions s i , i =1 , . . . , .Following Dzyaloshinsky, we des ribe the magneti ordering in terms of the following four ve tors: ~m = ~s + ~s + ~s + ~s ,~l = ~s − ~s − ~s + ~s ,~l = ~s − ~s + ~s − ~s ,~l = ~s + ~s − ~s − ~s . The ve tor ~m , the total magneti moment, orresponds to ferromagnetism (if it is the only non-zero ve tor, theordering is ferromagneti ), the ve tors ~l i (cid:21) to antiferromagnetism, ea h of them des ribing a parti ular pattern ofantiferromagneti ordering, [3℄. The Poisson bra kets for these variables, following from (1), read: { m α , m β } = ε αβγ m γ , { m α , l iβ } = ε αβγ l iγ , { l iα , l iβ } = ε αβγ m γ , { l iα , l jβ } = ε ijk ε αβγ l kγ , i = j, At the temperatures lose to that of the antiferromagneti transition the magneti energy an be expanded in powersof ~m, ~l i , as their values are small. Let us use the re tangular oordinate system with the z-axis dire ted along theaxis of the rystal, the x-axis (cid:21) along one of the se ond-order symmetry axes.The form of the possible terms in the magneti energy an be dedu ed from the analysis of the magneti symmetry.A thorough des ription of this approa h is given in [1℄. As is shown in [3℄, in the ase of α - F e O symmetry restri tionslead to the following general form of the se ond-order terms in the magneti energy of the system: E = A ~l + A ~l + A ~l + B ~m + α l z + α l z + α l z + b m z + β ( l x m y − l y m x ) + β ( l x l y − l y l x ) . Here the (cid:28)rst four terms orrespond to the ex hange intera tion, the other terms (cid:21) to the relativisti spin-latti eintera tion and the magneti dipolar intera tion. Experimental results show that in α - F e O the antiferromagneti stru ture orresponds to the ve tor ~l , [3℄, whi h means that in equilibrium approximately ~s = − ~s = − ~s = ~s .There is also weak ferromagnetism des ribed by the ve tor ~m . So, by physi al onsiderations, the system an berestri ted to only these two ve tors, whi h leads to the Poisson bra kets: { m α , m β } = ε αβγ m γ , { m α , l β } = ε αβγ l γ , { l α , l β } = ε αβγ m γ , (2)and the Hamiltonian, [3℄: H = A ~l + B ~m + α l z + b m z + β ( l x m y − l y m x ) + C ~l , (3)where ~l = ~l , the index 1 is omitted in all positions, and normalizing fa tors are introdu ed. It should be noted thatthis Hamiltonian has a forth-order term depending on the main magneti ve tor ~l = ~l . This model will be the obje tof our onsideration. III. SYMMETRY AND REDUCTIONIt is easy to noti e that the Hamiltonian (3) is invariant with respe t to the rotation about the z-axis, i.e. theanisotropy axis. This leads to m z being a (cid:28)rst integral. Indeed, al ulation shows that ˙ m z = { m z , H } = 0 . Moreover,the Poisson algebra (2) has two Casimir fun tions: ( ~m + ~l ) , ( ~m − ~l ) . To take advantage of these fa ts, let us introdu e the following variables: ~g = ~m + ~l , ~h = ~m − ~l , whi h are simply ~g = ~s + ~s , ~h = ~s + ~s in terms of the sublatti e magnetizations. Their Poisson bra kets have theusual form for angular momentum: { g α , g β } = ε αβγ g γ , { h α , h β } = ε αβγ h γ , { g α , h β } = 0 . The stru ture of the Poisson algebra given above admits of a redu tion that substantially simpli(cid:28)es investigatingthe dynami s of the system. The key point is to ast the Poisson bra kets for ve tor dynami al variables in a formthat relies on the s alar ones. The idea is essentially due to K. Pohlmeyer, [5℄, who employed it for studying thealgebra of urrents in (cid:28)eld theory. In paper [6℄ the method was used to study the spin dynami s in the B-phase ofsuper(cid:29)uid He in the regime of turned o(cid:27) magneti (cid:28)eld.In our ase the redu tion is obtained through the following system of variables: ~g = ( p g − g z cos ϕ g , p g − g z sin ϕ g , g z ) ,~h = ( p h − h z cos ϕ h , p h − h z sin ϕ h , h z ) . The new variables have the following sense: g (cid:21) the modulus of ~g , g z (cid:21) its z- omponent, and ϕ g (cid:21) its longitude angle(in xy-plane).Thus, we take into a ount the symmetry of the system and turn to the s alar quantities, ne essary for thePohlmeyer redu tion. It is worthwhile to note that a similar transformation was used in papers [8℄, [9℄ for studying haoti dynami s in two-spin systems.Using the reverse transformation: g = q g x + g y + g z , ϕ g = arctan g y g x , we obtain the following Poisson bra kets: { ϕ g , g z } = 1 , { g, g z } = 0 , { g, ϕ g } = 0 , { ϕ h , h z } = 1 , { h, h z } = 0 , { h, ϕ h } = 0 , all the g(cid:21)h bra kets being zero.Thus, we have got two advantages. The (cid:28)rst one is the lowering of the phase spa e dimensionality by two units, as g and h are Casimir fun tions, and their values an be (cid:28)xed. Therefore, we have a system with two pairs of anoni ally onjugate variables: ϕ g , g z and ϕ h , h z . The se ond one is that we an easily employ the fa t that m z = g z + h z is a(cid:28)rst integral. Indeed, another formulation of this is that the system is invariant with respe t to the rotation aboutthe z-axis. This means that the Hamiltonian should depend only on the di(cid:27)eren e of the angles ϕ g − ϕ h , not on theirspe i(cid:28) values. In fa t, al ulation of the Hamiltonian after the above transformations gives: H = 12 A h − p g − g z p h − h z cos ( ϕ g − ϕ h ) − g z h z + g + h i ++ B h p g − g z p h − h z cos ( ϕ g − ϕ h ) + 2 g z h z + g + h i ++ α ( g z − h z ) + b ( g z + h z ) − β p g − g z p h − h z sin ( ϕ g − ϕ h )++ C h − p g − g z p h − h z cos ( ϕ g − ϕ h ) − g z h z + g + h i . Applying the anoni al transformation: u = ϕ g − ϕ h √ , p u = g z − h z √ ,v = ϕ g + ϕ h √ , p v = g z + h z √ , we obtain the Hamiltonian: H = A (cid:16) − cos (cid:0) √ u (cid:1) p g − ( p u + p v ) p h − ( p u − p v ) + g + h + p u − p v (cid:17) + B (cid:16) cos (cid:0) √ u (cid:1) p g − ( p u + p v ) p h − ( p u − p v ) + g + h − p u + p v (cid:17) + αp u + bp v − β sin (cid:0) √ u (cid:1) p g − ( p u + p v ) p h − ( p u − p v ) + C (cid:16) − cos (cid:0) √ u (cid:1) p g − ( p u + p v ) p h − ( p u − p v ) + g + h + p u − p v (cid:17) . (4)It is easy to see that v does not enter this fun tion. So, as noted above, p v is a (cid:28)rst integral, and as the variables u, p u have zero Poisson bra kets with it, we on lude that the Hamiltonian 4 de(cid:28)nes a separate Hamiltonian systemin variables u, p u only. In other words, after (cid:28)xing the integrals of motion the fun tion 4 be omes a member of thePoisson subalgebra of dynami variables generated by u, p u and the (cid:28)rst integrals, thus de(cid:28)ning Hamiltonian dynami swithin this subalgebra.The study of the dynami s of our system then pro eeds as follows. Firstly, we study phase pi tures of the redu edsystem for u, p u . Se ondly, we onsider the "lift" of the obtained two-dimensional dynami s to the six-dimensionalspa e of the initial spin variables. Results obtained in this way admit of graphi representation.The dynami al regimes of the redu ed system an be studied by means of phase portraits in the u − p u phaseplane. A typi al portrait is shown in (cid:28)g. 1. It ontains several (cid:28)xed points: enters and saddles, and separatri es,ea h of whi h either onne ts a pairs of saddles or forms a loop having the origin and the end in the same saddle.A typi al traje tory going around a enter not far from it is shown in (cid:28)gs. 2, 3 in the spa es of the ve tors ~l , ~m respe tively; a traje tory lose to a separatrix (cid:21) in (cid:28)gs. 4, 5. It's easy to see that stationary points in the phaseportrait orrespond to horizontal ir les in the spa es of the magneti ve tors ~l , ~m , i.e. to ir ular pre ession of theseve tors. Other traje tories manifest more ompli ated behaviour, as an be seen in (cid:28)gs. 6, 7.An espe ially interesting example is a traje tory going very lose to a separatrix. This means that for a long period oftime it approa hes a saddle point, manifesting roughly ir ular pre ession of magnetizations, but at a ertain momentit starts moving away from that saddle performing some ompli ated dynami s before rea hing another almost-steadyregime of ir ular pre ession.The author is grateful to prof. V.L. Golo for onstant attention to this work.The author a knowledges prof. V.I. Mar henko for useful ommuni ations.The work was supported by the grants RFFI 09-02-00551, 09-03-00779.[1℄ A.F. Andreev and V.I. Mar henko, Symmetry and the ma ros opi dynami s of magneti materials, Sov. Phys. Usp., 23(1), pp. 21-34 (1980).[2℄ E.S. Borovik, V.V. Eremenko, A.S. Milner, Le tures on Magnetism, Physmathlit, Mos ow (2005).[3℄ I. Dzyaloshinsky, A thermodynami theory of (cid:16)weak(cid:17) ferromagnetism of antiferromagneti s, Journal of Physi s and Chem-istry of Solids, vol. 4, issue 4, pp. 241-255 (1957).[4℄ E.T. Whittaker, A Treatise on the Analyti al Dynami s of Parti les and Rigid Bodies, Cambridge University Press, 4thedition (1989).[5℄ K. Pohlmeyer, Integrable hamiltonian systems and intera tions through quadrati onstraints, Comm. math, phys., 46,207-221 (1976).[6℄ V.L. Golo, Nonlinear regimes in spin dynami s of super(cid:29)uid He, Letters in Mathemati al Physi s, Vol. 5, N. 2 (1981).[7℄ V.I. Mar henko, A.M. Tikhonov, On the NMR spe trum in antiferromagneti
CsMnI , JETP Lett., 69, p. 44 (1999).[8℄ N. Srivastava, C. Kaufman, G. M(cid:4)uller, R. Weber and H. Thomas, Integrable and nonintegrable lassi al spin lusters, Z.Phys. B 70, pp. 251-268 (1988).[9℄ D.T. Robb and L.E. Rei hl, Chaos in a two-spin system with applied magneti (cid:28)eld, Phys. Rev. E 57, N. 2, pp. 2458-2459(1998). - - - - p u FIG. 1: A typi al phase portrait of the system for the variables u, p u = l z / √ .FIG. 2: A traje tory surrounding a enter (ellipti (cid:28)xed point) in the phase portrait ((cid:28)g. 1), shown in the spa e of the oordinates of the antiferromagneti ve tor ~l . The urve is winding over a 2D-torus.FIG. 3: A traje tory surrounding a enter (ellipti (cid:28)xed point) in the phase portrait ((cid:28)g. 1), shown in the spa e of the oordinates of the ferromagneti ve tor ~m . The urve is on(cid:28)ned to a horizontal plane m z = const . - l x - l y - - l z FIG. 4: A traje tory lose to a separatrix onne ting two saddles in the phase portrait ((cid:28)g. 1), shown in the spa e of the oordinates of the antiferromagneti ve tor ~l . The ve tor migrates between two horizontal ir les orresponding to the twosaddles. - - m x - - m y m z FIG. 5: A traje tory lose to a separatrix onne ting two saddles in the phase portrait ((cid:28)g. 1), shown in the spa e of the oordinates of the ferromagneti ve tor ~m . The ve tor migrates between two ir les orresponding to the two saddles. The urve is on(cid:28)ned to a horizontal plane m z = const . - l x - l y - - l z FIG. 6: A traje tory taking S-shaped path between several enters and saddles in the phase portrait ((cid:28)g. 1), shown in thespa e of the oordinates of the antiferromagneti ve tor ~l . - m x - m y m z FIG. 7: A traje tory taking S-shaped path between several enters and saddles in the phase portrait ((cid:28)g. 1), shown in thespa e of the oordinates of the ferromagneti ve tor ~m . The urve is on(cid:28)ned to a horizontal plane m z = constconst