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Berry phase of magnons in textured ferromagnets

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Berry phase of magnons in textured ferromagnets
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  Berry phase of magnons in textured ferromagnets V. K. Dugaev, 1,2 P. Bruno, 1 B. Canals, 3 and C. Lacroix 3 1  Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2, 06120 Halle, Germany 2  Department of Physics and CFIF, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal 3  Laboratoire Louis Néel, CNRS, BP 166, 38042 Grenoble, Cedex 09, France  Received 4 March 2005; published 26 July 2005  We study the energy spectrum of magnons in a ferromagnet with topologically nontrivial magnetizationprofile. In the case of inhomogeneous magnetization corresponding to a metastable state of ferromagnet, thespin-wave equation of motion acquires a gauge potential leading to a Berry phase for the magnons propagatingalong a closed contour. The effect of magnetic anisotropy is crucial for the Berry phase: we show that theanisotropy suppresses its magnitude, which makes the Berry phase observable in some cases, similar to theAharonov-Bohm effect for electrons. For example, it can be observed in the interference of spin wavespropagating in mesoscopic rings. We discuss the effect of domain walls on the interference in ferromagneticrings, and propose some experiments with a certain geometry of magnetization. We also show that the non-vanishing average topological field acts on the magnons like a uniform magnetic field on electrons. It leads tothe quantization of the magnon spectrum in the topological field.DOI: 10.1103/PhysRevB.72.024456 PACS number  s  : 75.45.   j, 75.30.Ds, 75.75.  a I. INTRODUCTION The Berry phase theory 1–3 allowed generalization of theidea of Aharonov-Bohm effect 4 on electrons in the electro-magnetic potential, to an analogous effect related to a gaugepotential, which arises during the adiabatic motion of a quan-tum system in a parametric space. Up to now, much efforthas been directed to better understand and find an experi-mental confirmation for the Berry phase of electrons, forexample, in the case of electrons moving in a varying mag-netization field of the inhomogeneous ferromagnet. 5,6 One of the most intriguing consequences of the Berryphase theory is the possibility of the Aharonov-Bohm-typeeffect on electrically neutral particles or boson fields. 7,8 Anexample of the adiabatic phase for the polarized light hasbeen investigated by Pancharatnam 9 and Berry. 10 The otherexample is the Aharonov-Bohm effect for the exciton, 11 which is a bound state of an electron and a hole in semicon-ductors.Here, we consider the effect of the gauge potential andBerry phase on the propagation of magnons in textured fer-romagnets. Such quasiparticles are usually viewed as the el-ementary excitations of the ordered homogeneous state of aferromagnet, but they can also be used to classify the excitedstates near a metastable inhomogeneous magnetic configura-tion. These magnons describe the dynamics of weakly ex-cited inhomogeneous ferromagnet.The dynamics of magnetization in nanomagnets has beenthe focus of recent activity 12 because of its importance formagnetoelectronic applications. 13,14 It includes the switchingof magnetization by electric current, spin pumping, magne-tization reversal in microscopic spin valves, etc. Usually, themagnons play a negative role in the magnetization dynamicslimiting the frequency of magnetic reversal, and also leadingto the energy dissipation. However, they can probably beused in the spin-transport phenomena such as the spin cur-rents of magnetically polarized electrons.Here, we study the energy spectrum of spin waves in fer-romagnets with a static inhomogeneous magnetization pro-file, and we demonstrate the possibility of observation of theBerry phase in the interference experiments on spin waves inmagnetic nanostructures. Recent results of the micromag-netic computer simulation 15–17 of such systems demonstratethat the interference of spin waves actually can be observedin magnetic nanorings with domain walls.The equation for spin-wave excitations in a general caseof arbitrary local frame, depending on both coordinate andtime, was found long ago by Korenman  et al. 18 in the contextof local-band theory of itinerant magnetism. 19 Here, we usean idea of this method to relate the adiabatic space transfor-mation to the Berry phase, and to find corresponding prop-erties of the spin waves in a topologically nontrivial inhomo-geneous magnetic profile, which is a metastable state of theferromagnet. We show that the magnetic anisotropy is a cru-cial element determining the possibility of observation of theBerry phase in real experiments. II. MODEL AND SPIN WAVE EQUATIONS We consider the model of a ferromagnet described by theHamiltonian, which includes the exchange interaction, aniso-tropy, and the interaction with external magnetic field. It hasthe following general form:  H   = 12  d  3 r  a    n    r    r  i  2 + F   n  r   ,   1  where  n  r   is the unit vector oriented along the magnetiza-tion  M  r   at the point  r ,  a  is the constant of exchange inter-action, F   n  r   is a function determining the magnetic aniso-tropy   correspondingly, it includes a certain number of tensors relating the components of vector  n  r   and the de-pendence on external field, and  M  0  is the magnitude of mag-netization.Due to the condition  n 2  r  =1, the model is constrainedand belongs to the class of nonlinear     models. 20 The sta- PHYSICAL REVIEW B  72 , 024456   2005  1098-0121/2005/72  2   /024456  8   /$23.00 ©2005 The American Physical Society024456-1  tionary   saddle point   solutions for the magnetization vector n 0  r  , describing metastable states of the ferromagnet, can befound by minimizing Hamiltonian   1   with the constraint n 2  r  =1. It was shown   see, e.g., Refs. 21 and 22   that suchmetastable states with inhomogeneous magnetization profileare related to the topology of ferromagnetic ordering, andthey can include skyrmions, magnetic vortices, and other to-pological objects.We are interested in describing the dynamics of smalldeviations    n  r   from a certain metastable profile  n 0  r   witha nonuniform magnetization,  n  r  = n 0  r  +   n  r  ,     n  r   1. Correspondingly, we assume that the solution of asaddle-point equation describing the state  n 0  r   is alreadyknown.We perform a local transformation n ˜  r   = R  r  n  r  ,   2  using the orthogonal transformation matrix R  r  . By defini-tion, it determines the rotation of local frame in each point of the space, so that the magnetization in the local frame isoriented along the  z  axis,  n ˜ 0 =  0,0,1  . Then, we considersmall deviations of magnetization  s  r   from  n ˜ 0 . Since  s  r   issmall and vectors  n ˜ 0  are oriented along  z , the vectors  s  r   liein the  x  –  y  plane.The transformation matrix in Eq.   2   is taken in a generalform of orthogonal transformationR  r   =  e i    r   J   z e i    r   J   y e i    r   J   z ,   3  where    ,    ,     are the Euler angles determining an arbitraryrotation of the coordinate frame, and  J   x  ,  J   y , and  J   z  are thegenerators of 3D rotations around the  x  ,  y , and  z  axes, re-spectively.Two rotation parameters   for definiteness, the angles    and      can be used to define the frame with the  z  axis alongthe vector  n 0  r  . In the absence of anisotropy, the additionalrotation to the angle     is purely gauge transformation. How-ever, in a general case of anisotropic system, this rotationallows one to choose the local frame in correspondence withthe orientation of anisotropy axes.The Hamiltonian of exchange interaction   the first term inEq.   1   in the rotated frame has the following form:  H  ex  =  a 2  d  3 r    n ˜      r  i −  A i     n ˜     2 ,   4  where the gauge field  A i  r   is defined by  A i  r   =       r  i R  R −1 .   5  Transformation   3   and gauge potential   5   are 3  3 matricesacting on the magnetization vectors. The matrix  A i  r   canalso be presented as  A i  r   =  i A i    r   J    ,   6  where  A i    r   belongs to the adjoint representation of the ro-tation group.Using   3   and   6  , we find the explicit dependence of thegauge potential on the Euler angles A i x   r   = sin          r  i − sin     cos          r  i , A i y  r   = cos          r  i + sin     sin          r  i , A i z  r   =        r  i + cos          r  i .   7  The magnetic anisotropy described by the second term inthe right-hand part of    1   gives, after transformation to thelocal frame, a function  F  ˜   n ˜  r   with correspondingly trans-formed tensor fields. Here, we do not restrict the generalconsideration of the problem by any specific form of theanisotropy, but in the following we consider the most impor-tant examples of easy-plane- and easy-axis anisotropy.The Landau-Lifshitz equations for the magnetization inthe locally transformed frame are   n ˜      t  = −     M  0        n ˜         H    n ˜   −  i        H      i     n ˜      ,   8  where          is the unit antisymmetric tensor, and  i     =     r  i        −  A i      9  is the covariant derivative. The right-hand part of Eq.   8  vanishes for the magnetization profile corresponding to ametastable state. This is seen from the Landau-Lifshitz equa-tion in the unrotated srcinal frame. In the following, we willuse Eq.   8   for the small deviations of magnetization fromthe metastable state. Hence, we will consider in the right partof    8   only the terms linear in deviations.Using   1  ,   4  , and   8  , we find the equations for weakmagnetic excitations near the metastable state   spin waves    s  x    t  = −  c s    2 s  y   r  i 2  − 2 A i z   s  x    r  i −   A i z  2 s  y  −    A i z   r  i s  x   +   A i y  2 s  y + A i x  A i y s  x    +     M  0   2 F  ˜    n ˜   x    n ˜   y s  x   +     M  0   2 F  ˜    n ˜   y 2  s  y ,   10    s  y   t  =  c s    2 s  x    r  i 2  + 2 A i z   s  y   r  i −   A i z  2 s  x   +    A i z   r  i s  y  +   A i x   2 s  x  + A i x  A i y s  y   −     M  0   2 F  ˜    n ˜   x    n ˜   y s  y  −     M  0   2 F  ˜    n ˜   x  2  s  x  ,   11  where  c s =   a  /   M  0  is the stiffness.Using   10   and   11  , we can also present the equations forcircular components of the spin wave,  s ± = s  x  ± is  y DUGAEV  et al.  PHYSICAL REVIEW B  72 , 024456   2005  024456-2  ± i   s ±   t  =  −  c s      r  i  iA i z  2 −  V   r   +    2  M  0   2 F  ˜    n ˜   x  2 +    2  M  0   2 F  ˜    n ˜   y 2  s ±  +  −  w  r   −  ic s A i x  A i y +  i    M  0   2 F  ˜    n ˜   x    n ˜   y +    2  M  0   2 F  ˜    n ˜   x  2  −    2  M  0   2 F  ˜    n ˜   y 2  s  ,   12  where  V   r   and  w  r   are, respectively, the effective potentialand a mixing field acting on the spin wave V   r   =  c s 2   A i x   2 +   A i y  2  ,   13  w  r   =  c s 2   A i x   2 −   A i y  2  .   14  Equations   12   for  s +  r , t    and  s −  r , t    are complex conju-gates to each other since they both describe the same spinwave with real components  s  x   r , t    and  s  y  r , t   .We can see that  V   r   is an effective potential profile forthe propagation of spin wave. Due to the terms  w  r   and ic s A i x  A i y in   12  , the equations for circular components  s + and  s −  are coupled even in the absence of anisotropy. Allthese terms are of the second order in derivative of the rota-tion angle, and they are small in the adiabatic limit, corre-sponding to a smooth variation of the magnetization vector n  r  . III. SEMICLASSICAL APPROXIMATION Equations   10   and   11   can be solved in the semiclassicalapproximation. The condition of its applicability is a smoothvariation of gauge potential  A i    r   and fields related to theanisotropy, as well as the external magnetic field, at thewavelength of the spin wave,  kL  1, where  k   is the wavevector of the spin wave and  L  is the characteristic length of the variation of   A i    r   and  F   n  r   more exactly, the mini-mum of the corresponding characteristic lengths  . Note thatthe condition of applicability of the semiclassical approxima-tion to solve the spin-wave equations does not require anysmallness of the gauge potential itself.Starting from Eqs.   10   and   11  , we look for a generalsemiclassical solution in the form s  x   r , t    =  a  cos     r   −    t    +  b  sin     r   −    t   ,   15  s  y  r , t    =  d   sin     r   −    t    +  f   cos     r   −    t   ,   16  with arbitrary coefficients  a ,  b ,  d  ,  f  , and a smooth function    r  , so that we can neglect the second derivative of      r  over coordinate  r . Substituting   15   and   16   in   10   and   11  ,we can find four equation for the  a ,  b ,  d  ,  f   coefficients.The solution   15   and   16   describes the elliptic spin wavewith an arbitrary choice of the axes  x   and  y , and, generally,with a varying in space orientation of the principal axes of the ellipse. We can simplify our consideration by choosingthe angle     r   at each point of the space in accordance withthe orientation of the principal axes. The correspondingequation for     r   can be found from the condition of   b =  f  =0 in Eqs.   15   and   16  c s A i x  A i y −     M  0   2 F  ˜    n ˜   x    n ˜   y = 0.   17  Using   17   and neglecting the terms with derivative of   A i z ,which are small in the semiclassical approximation, we writethe spin-wave equations   10   and   11   as   s  x    t  = −  c s    2 s  y   r  i 2  − 2 A i z   s  x    r  i −    A i z  2 s  y  +   A i y  2 s  y   +     M  0   2 F  ˜    n ˜   y 2  s  y ,  18    s  y   t  =  c s    2 s  x    r  i 2  + 2 A i z   s  y   r  i −   A i z  2 s  x   +   A i x   2 s  x    −     M  0   2 F  ˜    n ˜   x  2  s  x  .  19  Note that by fixing the angle     in Eq.   17  , we are choosingthe gauge, which defines completely the potential A i   . We doit in spirit of the usual fixing gauge in the Wentzel-Kramers-Brillouin   WKB   approximation.After substitution of    15   and   16   with  b =  f  =0 into   18  and   19  , we come to the following equation for the momen-tum  k  i  r       r   /    r  i :     + 2 c s A i z k  i  2 −   c s  k  i 2 +   A i z  2 −   A i y  2   + 2  p  x   c s  k  i 2 +   A i z  2 −   A i x   2   + 2  p  y   = 0,   20  where  p  x  ,  y  r   =    2  M  0   2 F  ˜    n ˜   x  ,  y 2  ,   21  are the anisotropy parameters.Equation   20   should be solved for  k  i  r   as a function of smooth inhomogeneous field  A i    r  . This equation does notconstraint the orientation of   k  r   but determines the magni-tude of vector  k  r   for each direction in the momentumspace. Let us take vector  k  r   along an arbitrary direction,defined by a unity vector  g . Then, we can rewrite   20   as     + 2 c s kg i A i z  2 −   c s  k  i 2 +   A i z  2 −   A i x   2   + 2  p  x     c s  k  i 2 +   A i z  2 −   A i y  2   + 2  p  y   = 0,   22  and we come to the fourth-order algebraic equation for  k   r  .It can be solved numerically, and a resulting dependence of  k  i  r   on the gauge field in the integral     r  =  C  k  i  r  dr  i  leadsto the Berry phase acquired by the spin wave propagatingalong the contour  C  .We can find the solution of Eq.   22   analytically in thelimit of weak gauge potential   A i z   k  , which corresponds tothe adiabatic variation of the magnetization direction  n  r  and also the adiabatic rotation in space of the elliptic trajec-tory,        /    r  i   k  . Then, in the first order of   A i z we find BERRY PHASE OF MAGNONS IN TEXTURED FERROMAGNETS PHYSICAL REVIEW B  72 , 024456   2005  024456-3  k  i  r    g i   c s    2 +  p 2  1/2 −  p  1/2 +  g i g  j A  j z  r  1 +  p 2  /    2  −1/2 ,  23  where  p =   p  x  −  p  y  .Using Eq.   23   and taking the vector  g  along the tangentat each point of a closed contour  C  , we find the Berry phase    B  C    =  C  A i z  r  dr  i  1 +  p 2  /    2  1/2 .   24  As follows from   24  , the Berry phase     B  C    in the aniso-tropic system acquires an additional factor     1+  p 2  /    2  −1/2 depending on the magnetic anisotropy parameter  p .The denominator in   24   has a simple geometrical inter-pretation. Indeed, the coefficients  a  and  d   in the semiclassi-cal solution   15   and   16   are the ellipse parameters, whichare related to the anisotropy factor  pad  =   1 +  p 2   2  1/2 −  p   .   25  Correspondingly, we can relate the parameter    =  1+  p 2  /    2  −1/2 in Eq.   24   to the geometry parameters of theellipse   = sin 2  ,   26  where  =arctan  d   /  a  .Using definition   7  , the Berry phase finally can be pre-sented as    B  C    =  C      − 1       +      r  i +    cos    − 1        r  i  dr  i .  27  In this expression we extracted a term proportional to 2    N  ,  N   Z . This allows us to avoid the multivaluedness of Berryphase in the absence of anisotropy when    =1. 23 The firstterm in   27   is proportional to the total winding number of rotations associated with the angles     and    , whereas thesecond term is a spherical angle on  S  2 , which is the mappingspace of the vector field  n  r  . The second term in   27   has astandard interpretation of the Berry phase as the magneticflux penetrating the contour on  S  2 , when the field is createdby a monopole at the center of Berry sphere. Following thisidea, one can interpret the first term in   27   as the flux cre-ated by the magnetic string along the  z  axis, penetratingthrough the mapping contour on the unit circle. 23 In accor-dance with Eq.   27  , this contribution to the Berry phasevanishes for isotropic magnetic systems,    =1. The first termin   27   is the  topological  Berry phase   it depends only on thewinding number  , in contrast to the  geometric  Berry phase of the second term in   27  . 23 As follows from   24  , the effective gauge field for spinwaves in the anisotropic system is  A ˜  i =    A i z , and the corre-sponding topological field acting on the magnons can be cal-culated as the curvature of connection  A ˜  i  r   B i  =   ijk    A k  z   r   j +   ijk        r   j A k  z .   28  Note that there is a contribution related to the variation inspace of the anisotropy parameters   second term in Eq.   28  .We consider now in more detail the motion of elliptic spinwave in the adiabatic regime. The anisotropy suppresses oneof the components  s  x   or  s  y  breaking the symmetry with re-spect to rotations around the  z  axis. Correspondingly, there isno gauge invariance  s + → e i   s +  and  s − → e − i   s −  for the circu-lar components, and the motion of magnetization in the spinwave is elliptical. In the adiabatic limit of    A i z   k  , the solu-tions for  s  x   and  s  y  are given by Eqs.   15   and   16   with  b =  f  =0 and the ratio of amplitudes   a  /  d   . Thus, we couldexpect the local invariance to transformations preserving thevalue of   s  x  2 +  d   /  a  2 s  y 2 =const instead of simple rotations in the  x  –  y  plane.Using the Fourier transformation of Eqs.   18   and   19   for s  x  ,  y , we find the following equation for the elliptic compo-nents of spin wave,  s ˜  ± = s  x  ± i  d   /  a  s  y :     + 2 c s k  i A i z −  c ˜  s  k  i 2 +   A i z  2   −  pd  2 a  s ˜  + −  a 2 d   c s  k  i 2   +   A i z  2   1 −  d  2 a 2   −  pd  2 a 2   s ˜  −  = 0,  29  and the complex conjugate to   29  , where  c ˜  s = c s  1+ d  2  /  a 2  a  /2 d  , and we determine the  d   /  a  from the conditionof vanishing of the second bracket in Eq.   29  . This conditiondetermines the ellipticity factor, and we find that it coincideswith Eq.   25   in the limit of    A i z   k  . Thus, we come to thefollowing equation for the elliptic wave in the gauge field:     + 2 c s k  i A i z −  c ˜  s  k  i 2 +   A i z  2   −  pd  2 a  s ˜  +  = 0.   30  This equation is not gauge invariant but in the adiabatic re-gime, neglecting the difference in small terms of the order of   A i z  2 , we can present it as     −  c ˜  s  k  i  −   A i z  2 −  pd  2 a  s ˜  +  = 0.   31  Equation   31   contains a factor     1 before A i z , and formallylooks like the equation of motion of a particle moving in thereduced gauge field, which in turn leads to an effective sup-pression of the Berry phase. The calculation of Berry phaseusing Eq.   31   with the gauge field suppressed by factor    leads us again to Eq.   24  .In the absence of anisotropy and in the adiabatic approxi-mation, the solution of spin-wave equations has a simpleform. The equations for circular components   12   are sepa-rated± i   s ±  0    t  = −  c s      r  i  i A i z  2 s ±  0  ,   32  and the corresponding solution is DUGAEV  et al.  PHYSICAL REVIEW B  72 , 024456   2005  024456-4  s +  r , t     exp  i  k  i  r  dr  i  −  i   t   ,   33  with  k  i  r  g i    /  c s 1/2 + g i g  j A  j z  r  . The spin wave propagatingalong a closed contour  C   acquires the Berry phase of Eq.  24   with  p =0.Using Eqs.   7   we can present the topological field   28   inthe absence of anisotropy as  B i  = − sin    ijk        r   j       r  k  .   34  It does not depend on the angle    , related to the choice of gauge as in the case of electromagnetism.By creating a certain metastable configuration of the mag-netization  n 0  r   in the ferromagnet, we simulate an effectivegauge potential A ˜  i  r  , acting on the spin waves similar to themagnetic field in case of electrons. In particular, when theaveraged in space topological field   28   is not zero, therearises the Landau quantization of the energy spectrum of magnons. In the absence of anisotropy, we find the quantizedspectrum    n =2  c s  B  n +1/2  , where   …   means the av-erage in space. IV. INTERFERENCE OF SPIN WAVES IN MESOSCOPICRINGS Let us consider now the ring geometry of a ferromagnetwith a topologically nontrivial metastable magnetization n 0  r  . It can be, for example, a magnetization vortex   Fig.1  a   or an even number of domain walls in one branch of thering as presented in Fig. 1  b  . Such a magnetization profilepresents a metastable magnetic state.Let us consider first the case when there is no anisotropy.If   kL  1   adiabatic regime  , the low-energy magnetic exci-tations of the metastable state are described by Eq.   32  . Dueto the presence of gauge potential  A i z  r  , there is a phaseshift of waves propagating from the point  A , where thewaves are excited, to the observation point  B   see Fig. 1  .The phase shift   Berry phase   equals the integral   A i z  r  dr  i along the ring, and by using the Stokes theorem can be cal-culated as the flux    of topological field  B  defined in Eq.  34  . It can also be presented as the spherical angle enclosedby the mapping of the ring to the circle at the unit sphere  S  2 .This way we can find the phase shift of 2    and     N  d   for Figs.1  a   and 1  b  , respectively, where an even  N  d   is the numberof domain walls in the right arm of the ring. For example, inthe case of Fig. 1  b   with two domain walls in the right arm,there is no interference of spin waves excited in  A  and com-ing to the point  B  because the corresponding phase shift is2   .In the absence of anisotropy, the interference in the ringcan be induced by rotating all magnetic moments from theplane to a certain angle      the corresponding mapping ispresented in Fig. 2  b  . The Berry phase associated with thepath along the ring will be smaller than 2   . For    =    /6 theBerry phase turns out to be    . This means that the experi-ment with interference of spin waves propagating from  A  to  B  through two different arms of the ring would result in acomplete suppression of the outgoing from  B  spin wave.Physically, it can be realized using the ring with very smalleasy-plane anisotropy,  p  /     1, in a weak external magneticfield along  z  axis.Asimilar idea was recently proposed by Schütz  et al. 24 forthe radial orientation of magnetic moments under inhomoge-neous magnetic field directed from some point at the axis of the ring. This magnetic field creates a “crown” of magneticmoments, and the corresponding mapping is similar to thatpresented in Fig. 2  b  .However, in the case of nonvanishing easy-plane aniso-tropy, there is no need to apply magnetic field to provide theinterference of spin waves propagating in the geometry of Fig. 1  b  . In this case the Berry phase is given by Eq.   27  with  p =     /2  M  0 =const, and we obtain the difference inphases for two waves     B =    N  d   /   1+  p 2  /    2  1/2 . Thus, the in-terference of spin waves should be clearly seen for the two-arm geometry with domain walls. The computer simulationexperiments 17 confirm this expectation.Another possibility to observe the interference of spinwaves can be presented by the geometry of a wide ring   thin-wall cylinder   as presented in Fig. 3. Assuming the easy-plane anisotropy of the ribbon, we obtain the ground statewith a homogeneous magnetization along the axis of cylin-der. The anisotropy axis is oriented radially in each point of the cylinder, and the corresponding local frame is shown inthe figure as  x    y   z  . Due to the homogeneous magnetization,we get    =0, and from   7   we obtain A i z  r  =      +     /    r  i . Thecomponents of anisotropy vector are  u  x  =cos     and  u  y =sin    . Using Eq.   34  , we find that the condition  u ˜   y =0 re-duces to    +   =   , and from   34   we obtain  u ˜   x  =1. Corre- FIG. 1. Two rings with topologically nontrivial magnetizationfield.FIG. 2. Mapping of the ring to the  n -space  S  2   red contour   inthe case of in-plane vortex magnetization shown in Fig. 1  a   a   andfor the same geometry with magnetization vector deviating from theplane to the angle      b  . The Berry phase is 2    in case   a   and     for   =    /6 in case   b  .BERRY PHASE OF MAGNONS IN TEXTURED FERROMAGNETS PHYSICAL REVIEW B  72 , 024456   2005  024456-5
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