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The variational symmetries and conservation laws in classical theory of Heisenberg (anti)ferromagnet
14 January 2002
Physics Letters A 292 (2002) 325–334 www.elsevier.com/locate/pla
The variational symmetries and conservation laws in classical theory of Heisenberg (anti)ferromagnet R.F. Egorov, I.G. Bostrem, A.S. Ovchinnikov ∗ Department of Theoretical Physics, Ural State University, Lenin Ave. 51, Ekaterinburg 620083, Russia Received 4 June 2001; received in revised form 26 October 2001; accepted 3 December 2001 Communicated by A.R. Bishop
Abstract The non-linear partial differential equations describing the spin dynamics of Heisenberg ferro- and antiferromagnet are studied by Lie transformation group method. The generators of the admitted variational Lie symmetry groups are derived and conservation laws for the conserved currents are found via Noether’s theorem. 2002 Elsevier Science B.V. All rights reserved. PACS: 02.20.T; 75.10.J Keywords: Symmetry groups; Noether theorem; Heisenberg model
1. Introduction As it has been demonstrated previously in a number of works the symmetry methods are very efficient tool in application to differential equations in physics. A subject of an especial interest is a study of invariance properties (symmetries) of the equations with respect to local Lie groups point transformations of dependent and independent variables. The detailed discussion and applications of the symmetries of differential equations and variational problems may be found in Refs. [1–3]. The non-linear dynamics of n-dimensional (anti)ferromagnet is described by non-linear second-order partial differential equations (PDE). In our previous work it has been shown that the system of differential equations are the Euler–Lagrange equations of a certain action functional and the corresponding long-wave action has been constructed via Vainberg’s theorem (see also Appendix A). It is well known that a part of one parameter symmetry groups of these equations turns out to be their variational symmetries as well, i.e., a symmetries of the associated action functional. According to Noether theorem [1] such an invariance of the elementary action is a necessary and sufficient condition of an existence of conservation laws for the smooth solutions of the initial PDE. The layout of the Letter is as follows. By using the Noether identity we define the variational symmetries among the point Lie symmetries of the initial differential equations of a 2D-dimensional ferromagnet. After they are * Corresponding author.
E-mail address: [email protected] (A.S. Ovchinnikov). 0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 8 1 3 - 1
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deduced we establish explicit formulae for the conserved densities and density currents involved in the conservation laws both for a Heisenberg ferromagnet and antiferromagnet. Apparently, the analysis presented may be easily generalized for an arbitrary dimension n 3.
2. Dynamical invariants of 2D ferromagnet The non-linear partial differential equations in three independent variables (x 1 , x 2 , x 3 ) = (t, r) and two dependent variables u1 = θ (t, r) and u2 = ϕ(t, r) describing the dynamics of Heisenberg ferromagnet in continuum approximation can be written as follows [4]: 2 F1 ≡ −S sin θ ϕt + β ∆θ − cos θ sin θ ∇ϕ = 0, 2 2 F ≡ S sin θ θt + β 2 cos θ sin θ ∇θ ∇ϕ + sin θ ∆ϕ = 0, (1) where β = J S 2 /h¯ . Here and throughout the following notations are taken: α have the range 1, 2 meaning θ and ϕ, correspondingly, the usual summation convention over a repeated index is employed, θj1 ...jk (or ϕj1 ...jk ) denote the kth-order partial derivatives of the dependent variables: ∂kθ . ∂x j1 ∂x j2 . . . ∂x jk The J denotes a multi-index J = (j1 , j2 , . . . , jk ) with jk = 1, . . . , 3 (k 0) pointing to the independent variables with respect to which one differentiate. Consider a local one-parameter Lie group of point transformations acting on a space Ω of the independent and dependent variables involved in the basic equations (1). The infinitesimal generator of such a group is a vector field Xˆ on Ω, θ(J ) = θj1 ...jk =
∂ ∂ ∂ + η2 , Xˆ = ξ k k + η1 ∂x ∂θ ∂ϕ
(2)
whose components ξ k and ηα are supposed to be functions of class C∞ on Ω. The infinitesimal invariance criterion of the equations F = (F1 , F2 ) under the group G is given by the determining equations Yˆ F F=0 = 0, (3) α Yˆ = Dˆ (J ) W ∂uα(J ) , (4) where Dˆ k means total differentiation, Dˆ k = ∂k + uαk(J ) ∂uα(J ) ,
(5)
ˆ and W α = ηα − ξ i uαi are the characteristics of the canonical vector field Yˆ which is equivalent to X: Yˆ = Xˆ − ξ k Dˆ k .
(6)
The determining equations (3) are realized as an over-determined system of linear homogeneous partial differential equations with respect to the unknown functions ξ k and ηα . In the determining equations the uα(J ) up to second-order derivatives are the independent variables and these equations must be fulfilled identically over them on the manifold F. A sufficient condition of an existence of conservation laws is an invariance of the elementary action associated with the Euler equations under a symmetry group of these equations. The starting point is the so-called Noether identity [1]: Yˆ + Dˆ i · ξ i = W α Eˆ α + Dˆ i Nˆ i .
(7)
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In Eq. (7) Nˆ i are the Noether operators given by the expressions Nˆ i = ξ i + Dˆ (s) W α (−1)r Dˆ (r) ∂uα(s)(r) ,
327
(8)
and Eˆ α = (−1)s Dˆ (s) ∂uα(s)
(9)
are the Euler–Lagrange operators. The variational symmetries on the Euler–Lagrange equations may be found via the Noether identity Yˆ (L) + Dˆ i · ξ i L = 0 (10) with the Lagrangian density L [5]:
1 2 2 1 2 L = L(θ, ϕt , θk , ϕk ) = S(cos θ − 1)ϕt − β sin θ ∇ϕ + ∇θ . 2 2
The dot in Eq. (7) means the differentiation rule Dˆ i · ξ i (L) ≡ LDˆ i (ξ i ) + ξ i Dˆ i (L). Substituting (11) into (10) one obtains Yˆ (L) = W 1 ∂θ L + Dˆ t W 2 ∂ϕt L + Dˆ k W 1 ∂θk L + Dˆ k W 2 ∂ϕk L,
(11)
(12)
where 2 β sin 2θ ∇ϕ , ∂ϕt L = S(cos θ − 1), 2 ∂ϕk L = −β sin2 θ ϕk .
∂θ L = −S sin θ ϕt − ∂θk L = −βθk ,
After a little manipulation one gets the following equation for the unknowns ξ k and ηα : 2 β 1 + S(cos θ − 1) Dˆ t η2 − ϕt Dˆ t ξ t − ϕk Dˆ t ξ k η −S sin θ ϕt − sin 2θ ∇ϕ 2 1 ˆ − βθk Dk η − θt Dˆ k ξ t − θl Dˆ k ξ l − β sin2 θ ϕk Dˆ k η2 − ϕt Dˆ k ξ t − ϕl Dˆ k ξ l 2 β 2 t β + S(cos θ − 1)ϕt − sin2 θ ∇ϕ Dˆ t ξ + Dˆ k ξ k = 0. − ∇θ 2 2
(13)
(14)
Hereinafter, we keep the upper index k just for the space coordinates, the corresponding time component will be written explicitly. The coefficients at the derivatives of the functions θ and ϕ constitute an over-determined system of linear homogeneous partial differential equations with respect to the unknowns ξ i and ηα . For example, extracting the terms at the independent variable θt and setting equal them to zero, 2 1 2 t 1 S ηθ2 − ϕk ξθk (cos θ − 1) + βθk ξkt + θk ξθt + ϕk ξϕt + β − sin2 θ ∇ϕ (15) − ∇θ ξθ = 0, 2 2 one obtains the conditions ηθ2 = ξθk = ξkt = ξϕt = ξθt = 0
(16)
that immediately imply ξ t = ξ t (t). The results of the full analysis of Eq. (14) are summarized in Table 1. Finally, we get ξ t = ξ t (t), ξ k = ξ k (r ), η1 = η1 (t, θ ), η2 = η2 (ϕ). The unknowns ξ k satisfy to Killing’s equation ξkl + ξlk = σ δlk . As is well known, it has a solution for an arbitrary σ just for n 3. However, for the system considered there is a solution in 2D case. Indeed, from Table 1 one have σ = 2ξkk = 2ηθ1 + ξtt + div ξ ,
σ = 2ξkk = 2η1 cot θ + ξtt + div ξ + 2ηϕ2
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Table 1 Equations for the unknowns ξ k and ηα Independent variable
Equation ηt2 = 0 (cos θ − 1)(ηϕ2 + div ξ ) = η1 sin θ −S(cos θ − 1)ξtk − β sin2 θ ηk2 = 0 ⇒ ξtk = ηk2 = 0 ηk1 = 0 ξ k = η1 cot θ + 1 (ξtt + div ξ ) + ηϕ2
0 ϕt ϕk θk 2 (∇ϕ)
k
2
sin2 θ (ξkl + ξlk ) = 0 ⇒ ξkl + ξlk = 0 ξkk = ηθ1 + 12 (ξtt + div ξ )
ϕk ϕl (k = l) )2 (∇θ
ξkl + ξlk = 0
θk θl (k = l)
ηϕ1 = 0
θk ϕk (∇ϕ)2 ϕk
ξϕk = 0
Table 2 Conservation laws
1 2 3 4
(no summation over k), that imply ξ = ξ (t), t
t
ξ r = a + Bˆ r,
Conservation law 2 + β (∇θ )2 = Dˆ k (βθt θk + βϕt ϕk sin2 θ ) Dˆ t β2 sin2 θ (∇ϕ) 2 Dˆ t (S(cos θ − 1)ϕk ) = Dˆ l (Tkl ) l ) = Dˆ k (& lmn x m Tnk ) (l = z) Dˆ t ([r × S(cos θ − 1)∇ϕ] Dˆ t (S(cos θ − 1)) = Dˆ k (βϕk sin2 θ )
0 T ˆ ˆ B = −B = − b
−b , 0
η1 = 0,
η2 = υ.
− b[r × ∇θ ]z The solution found results in the following expressions for the characteristics: W 1 = −ξ t θt − a ∇θ 2 t and W = υ − ξ ϕt − a ∇ϕ − b[r × ∇ϕ]z . One can derive corresponding conservation laws through Noether’s theorem procedure, Dˆ i Nˆ i (L) = 0,
(17)
under the condition Eˆ α L = 0. The Noether operators modify the Lagrangian density into a conserved quantity C i = Nˆ i (L) with a zero total divergence. Using the found variational symmetries we first construct the operators Nˆ i from formulae (8) and then calculate from (17) the corresponding conservation laws. The results are brought together in Table 2. The physical interpretation of the obtained relations is obvious. (1) The symmetry group of translations in time t → t + a yields to the energy conservation law with the 2 + (J S 2 /2)(∇θ )2 and the energy density current T = −(J S 2 θt ∇θ + energy density W = (J S 2 /2) sin2 θ (∇ϕ) 2 2 J S ϕt ∇ϕ sin θ ). and the canonical energy-momentum tensor (2) This law connects the momentum P = h¯ S(1 − cos θ )∇ϕ 2 Tkl = hLδ ¯ kl + hβθ ¯ k θl + h¯ βϕk ϕl sin θ,
(18)
and reflects the variational symmetry under space translations xl → xl + al . = [r × P ] through the angular momenta tensor under the space (3) The conservation of angular momenta L rotation around z-axis. (4) The conservation law corresponding to the transformations ϕ → ϕ + χ connects the scalar variable sin2 θ . N = S(1 − cos θ ) with the current j = β ∇ϕ
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It is useful to apply the obtained laws to some types of solutions of Eqs. (1). We restrict consideration by finite-amplitude spin-waves, dynamical soliton and skyrmion. r − ωt [6] with the dispersion h¯ ω = J S k2 cos θ0 . (1) The finite-amplitude spin-waves θ = θ0 = const and ϕ = k The last conservation law may be interpreted as a conservation of the magnon density N = S(1 − cos θ0 ). The corresponding magnon density current is j = (J S 2 /h¯ )k sin2 θ0 . The space components of the energy-momentum tensor are 2 S(1 − cos θ0 )h¯ ω + J 2S sin2 θ0 (kx2 − ky2 ) J S 2 sin2 θ0 kx ky Tˆkl = . 2 J S 2 sin2 θ0 kx ky S(1 − cos θ0 )h¯ ω − J 2S sin2 θ0 (kx2 − ky2 ) The momentum P = S(1 − cos θ0 )h¯ k is a times of the magnon density and the elementary momentum h¯ k. 2 2 2 The energy density W = (1/2)J S k sin θ0 ≈ N hω ¯ (θ0 1) has the corresponding energy density current sin2 θ0 . We note a non-additive character of energy density at finite values θ0 . T = J S 2 kω (2) For a dynamical soliton with axial symmetry one have to use the following parametrization: θ = θ (r) and ϕ = ωt [6]. The magnon density N = S(1 − cos θ (r)) has the zero magnon density current j = 0. The energymomentum tensor is 2 2 2 J S2 sin(2ωt) dθ −S(1 − cos θ (r))h¯ ω + J 2S cos(2ωt) dθ dr 2 dr Tˆkl = dθ 2 dθ 2 . J S2 J S2 sin(2ωt) −S(1 − cos θ (r)) h ω − cos(2ωt) ¯ 2 dr 2 dr The momentum P is zero, the energy density W = (J S 2 /2)(dθ/dr)2 and the energy density current T = 0. (3) For the skyrmion the following parametrization θ = 2 tan−1 (R/r) and ϕ = tan−1 (y/x) [7] is used (for simplicity we consider an unit topological charge). Then, N = 2S
r2
R2 , + R2
jr = 0,
jϕ =
J S 2 4R 2 r . h¯ (R 2 + r 2 )2
The momentum in polar coordinates of xy-plane Pr = 0,
Pϕ =
2h¯ S R 2 , r r 2 + R2
however, Tˆkl = 0. The energy density J L2z R2 = (R 2 + r 2 )2 h¯ 2 R 2 and the energy density current T = 0. W = 4J S 2
3. Dynamical invariants of 2D two-sublattice magnet In this section Lie transformation group methods will be applied to the partial differential equations describing the dynamics of 2D generic antiferromagnet (two-sublattice magnet). The established variational Lie symmetries for ferromagnet will be employed to derive antiferromagnet conservation laws revealing its important features. The action of 2D two-sublattice magnet is [5] (see also Appendix B) L=S
2
(cos θi − 1)ϕit − β −2 sin θ1 sin θ2 cos(ϕ1 − ϕ2 ) − 2 cos θ1 cos θ2 i=1
2 − sin θ1 cos θ2 sin(ϕ1 − ϕ2 ) ∇ϕ 2 1 ∇θ 1 ∇θ + cos θ1 cos θ2 cos(ϕ1 − ϕ2 ) ∇θ
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1 ∇ϕ 1 ∇ϕ 2 + sin θ1 sin θ2 cos(ϕ1 − ϕ2 ) ∇ϕ 2 + cos θ1 sin θ2 sin(ϕ1 − ϕ2 ) ∇θ 1 ∇θ 2 . + sin θ1 sin θ2 ∇θ
(19)
The time symmetry group t → t + a under which the elementary action is invariant has the infinitesimal symmetry generator Xˆ = ∂/∂t and Noether’s operators Nˆ t = 1 −
2 i=1
∂ ∂ − ϕit , ∂θit ∂ϕit 2
θit
Nˆ k = −
i=1
2 i=1
∂ ∂ − ϕit . ∂θ ik ∂ϕik 2
θit
i=1
By introducing the functions
C t = Nˆ t (L) = −β −2 sin θ1 sin θ2 cos(ϕ1 − ϕ2 ) − 2 cos θ1 cos θ2 1 ∇θ 1 ∇θ 2 − sin θ1 cos θ2 sin(ϕ1 − ϕ2 ) ∇ϕ 2 + cos θ1 cos θ2 cos(ϕ1 − ϕ2 ) ∇θ 1 ∇ϕ 1 ∇ϕ 2 + sin θ1 sin θ2 cos(ϕ1 − ϕ2 ) ∇ϕ 2 + cos θ1 sin θ2 sin(ϕ1 − ϕ2 ) ∇θ 1 ∇θ 2 + sin θ1 sin θ2 ∇θ
(20)
and C k = Nˆ k (L) = β cos θ1 cos θ2 cos(ϕ1 − ϕ2 ) + sin θ1 sin θ2 (θ1t θ2k + θ2t θ1k ) + β sin θ1 sin θ2 cos(ϕ1 − ϕ2 )(ϕ1t ϕ2k + ϕ2t ϕ1k ) + β cos θ1 sin θ2 sin(ϕ1 − ϕ2 )(θ1t ϕ2k + θ1k ϕ2t ) − β sin θ1 cos θ2 sin(ϕ1 − ϕ2 )(θ2t ϕ1k + θ2k ϕ1t )
(21)
one gets the energy conservation law Dˆ t (C t ) + Dˆ k (C k ) = 0. By the same manner one can obtain the generators and Noether’s operators of the translation group xk → xk + ak : ∂ Xˆ = k , ∂x
Nˆ t = −
2
∂ ∂ − ϕik , ∂θ it ∂ϕit 2
θik
i=1
i=1
Nˆ l = δkl −
2
∂ ∂ − ϕj k . ∂θj l ∂ϕj l 2
θj k
i=1
i=1
The momentum conservation law is 2
Dˆ t S(cos θi − 1)ϕik = Dˆ l (Tkl ),
(22)
i=1
with the tensor Tkl = Lδkl −
2 j =1
θj k ∂θjk L −
2
ϕj k ∂ϕjk L.
(23)
j =1
The conservation of angular momenta is 2 j =1
j = Dˆ k & znl x n Tlk , Dˆ t r × S(cos θj − 1)∇ϕ z
∂ ∂ Nˆ t = −& znl x n θlα α − & znl x n ϕlα α , ∂θt ∂ϕt
∂ Xˆ = & znl x n l , ∂x
∂ ∂ Nˆ k = & znk x n − & znl x n θlα α − & znl x n ϕlα α . ∂θk ∂ϕk
(24)
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The action is also invariant under a simultaneous rotation in both sublattices ϕ1 = ϕ1 + χ , ϕ2 = ϕ2 + χ . The generator and Noether operators are Xˆ =
2 ∂ , ∂ϕj
Nˆ t =
j =1
2 ∂ , ∂ϕj t
Nˆ k =
j =1
2 ∂ ∂ϕj k j =1
and they imply the conservation law 2
Dˆ t (S cos θj ) = β Dˆ k sin θ1 sin θ2 cos(ϕ1 − ϕ2 )(ϕ1k + ϕ2k )
j =1
− sin(ϕ1 − ϕ2 )(θ2k sin θ1 cos θ2 − θ1k sin θ2 cos θ1 ) .
(25)
4. Conclusions Lie transformation group methods have been applied to the partial differential equations describing the dynamics of ferro- and antiferromagnets and the variational symmetries of the action functional on the Euler–Lagrange equations are established. Each such symmetry gives rise to a conservation law with a conserved current. The explicit expressions for the conserved currents are found.
Acknowledgements This work was supported by the grant NREC-005 of US CRDF (Civilian Research and Development Foundation).
Appendix A Let us take the following notations: β (1) the matrix of Frechet derivatives fˆβα = s (∂f α /∂u(s))Dˆ (s) , where the sum is taken over all multi-indices s = (s1 , s2 , . . . , sk ) with sk = 1, . . . , 3 (k 0); (2) the adjoint operator of Frechet derivatives [2] β ∗ −Dˆ fˆα =
(s)
s
·
∂f α β
.
∂u(s)
One has to found a Lagrangian density L(x, u, u , . . .) that the initial set of differential equations are the Euler– Lagrange equations of the following action: S[u] =
t1
L(x, u, u , . . .) dx ≡
dt
dV L(x, u, u , . . .).
t2
From the ratio Eˆ α L = f α +
1 0
fˆ † − fˆ
α β
λuβ (x) dλ
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is seen that that sufficient condition of the solution of the problem is the self-adjoint condition of the fˆ-operator, ∂f α β
∂u(s)
∂f β s = (−1)s+r Cs+r , Dˆ (r) α ∂u(s)(r)
β or in the matrix notations fˆβα = (fˆ † )αβ = (fˆα )∗ . The straight calculation yields to
2 cos 2θ + β∆ −Sϕt cos θ − β(∇ϕ)
fˆ = Sθt cos θ + 2β cos 2θ (∇ϕ ∇θ ) + β sin 2θ ∆ϕ + S sin θ ∂t + β sin 2θ (∇ϕ ∇) and
∇) −S sin θ ∂t − β sin 2θ (∇ϕ ∇) + β sin2 θ ∆ β sin 2θ (∇ϕ
∇θ ) + β sin 2θ ∆ϕ Sθt cos θ + 2β cos 2θ (∇ϕ ∇) + S sin θ ∂t + β sin 2θ (∇ϕ . 2 β sin 2θ (∇ϕ ∇) + β sin θ ∆
2 −Sϕt cos θ − β(∇ϕ) cos 2θ + β∆ ∗ ˆ f = ∇) −S sin θ ∂t − β sin 2θ (∇ϕ
Obviously, the claimed condition is fulfilled. The Lagrangian density can be constructed via the homotopy formula [2] L[u] =
α
1 α
u (x)
f α (x, λu, λu , . . .) dλ.
0
Using the explicit form of differential equations it results in 1 L = −S(θ ϕt − ϕθt )
1 dλ λ sin λθ + βθ ∆θ
0
dλ λ 0
1 1 θ 2 2 dλ λ sin 2λθ + βϕ∆ϕ dλ λ sin2 λθ, + β ϕ ∇ϕ ∇θ − ∇ϕ 2 0
0
or, after some manipulations, in 2 β 2 β L = S(cos θ − 1)ϕt − sin2 θ ∇ϕ − ∇θ + S Dˆ t 2 2
β sin θ β ˆ ϕ + Dk θ ∇θ + 2 C0 ϕ ∇ϕ , 1− θ 2 θ
where
1 θ (θ − sin 2θ ) + sin2 θ . 4 The last two terms present total derivatives and may be dropped. C0 =
Appendix B In this appendix we show that the action employing the Lagrangian density of Eq. (19) takes the form of the action for a (d + 1)-dimensional non-linear sigma model [8–10]. The last model yields a long-wavelength, large S description of the dynamic of d-dimensional antiferromagnets in the vicinity of a ground state with collinear long-range Neel state. A more detailed discussion of the problem may be found, for example, in [11].
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We start with the long-wave action functional of a d-dimensional hypercubic lattice with a nearest-neighbour exchange interaction J < 0 constructed by using non-linear partial differential equations of motion of the individual spins [5], F = −S
2
t2 d r
i=1
1 dt
t1
t2 ∂ N i ∂ Ni 1 × dλ Ni − d r dt Ω|H |Ω, ∂λ ∂t h¯ t1
0
where the density of the energy is given as follows:
Ω|H |Ω = J S 2 −d sin Θ1 sin Θ2 cos(ϕ1 − ϕ2 ) − d cos Θ1 cos Θ2 1 ∇Θ 1 ∇Θ 2 − sin Θ1 cos Θ2 sin(ϕ1 − ϕ2 ) ∇ϕ 2 + cos Θ1 cos Θ2 cos(ϕ1 − ϕ2 ) ∇Θ 2 + sin Θ1 sin Θ2 cos(ϕ1 − ϕ2 ) ∇ϕ 2 1 ∇ϕ 1 ∇ϕ + cos Θ1 sin Θ2 sin(ϕ1 − ϕ2 ) ∇Θ 1 ∇Θ 2 . + sin Θ1 sin Θ2 ∇Θ Using the vectors N i (λ, t) = {sin λΘi cos ϕi , sin λΘi sin ϕi , cos λΘi }, N i (t) = Ni (1, t), the initial functional can be presented as F = −S
2
t2 d r
i=1
1 dt
t1
t2 ∂ N i J S2 ∂ Ni α α − × ∇N1 ∇N2 . dλ Ni d r dt −d N1 N2 + ∂λ ∂t h¯ α=x,y,z t1
0
In the semiclassical limit the spins will be predominantly oriented in opposite directions on the two sublattices. It allows to introduce the parametrization d 2d 2 2 + ad L n 1 − a 2d L N1 = n 1 − a L + a L, N2 = − via a continuum field of unit length n(r , t) which describes the local orientation of the Neel ordering. In the vicinity of the Neel order the n varies slowly on the scale of a lattice spacing a. The relationships include a component r , t) of the spins which is perpendicular to the local orientation of the Neel order. The continuum fields n and L L( obey the relationships = 0, n L
n 2 = 1,
2 a −2d L
satisfy the same boundary conditions in t as N (the lattice unit a which holds for all values of t and λ, and n , L will be taken unity hereinafter). By inserting the decompositions for N 1,2 in Ω|H |Ω and expanding the result in one obtains gradients and powers of L 2 2 . n − 2d L Ω|H |Ω ≈ J S 2 d − ∇ (B.1) The first term dJ S 2 presents a total derivative and may be dropped in further consideration. The Berry phase terms FB = −S
2
t2 d r
i=1
1 dt
t1
t2 2 i ∂ N ∂ϕi ∂ N i × dλ N i d r dt (cos Θi − 1) =S ∂λ ∂t ∂t
0
i=1
t1
fields: take the following form in n and L FB ≈ −2S
t2 d r
1 dt
t1
0
∂ L ∂ n ∂ n ∂ L × × dλ n + n , ∂λ ∂t ∂λ ∂t
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R.F. Egorov et al. / Physics Letters A 292 (2002) 325–334
are retained. The terms with ( where the terms of linear order in L n[∂ n /∂λ × ∂ n/∂t]) arising from the both sublattices compensate each other. It is valid for smooth spin configurations in the vicinity of Neel state. For nontrivial topological configurations this sum may be non-zero, hence the (Θ, ϕ)-representation used for the derivation of the conservation laws is more appropriate. ∂ n /∂t and ∂ n /∂λ are all perpendicular to n and have a zero triple The FB can be simplified since the vectors L, product: t2
FB = −2S
d r
1 dt
t1
∂ ∂ ∂ n ∂ n ×L + dλ n n L × . ∂t ∂λ ∂λ ∂t
(B.2)
0
The total t derivative gives zero and the total λ derivative yields a surface contribution at λ = 1. Putting together (B.1) and (B.2) one obtains the following functional: t2 2 ∂ n 1 2 2 × n + χ⊥ S L + ρs ∇ F =− d r dt 2S h¯ n L ∂t h¯ t1
with the spin-wave stiffness ρs = |J |S 2 and the transverse susceptibility χ⊥ = 2d|J |. If to replace t → iτ , the common 1/h¯ → β and multiply the functional by −i (then F → −βSAFM ), one obtains the following continuum limit path-integral for the antiferromagnet partition function: Z=
exp(−βSAFM ), D n D L
SAFM =
β d r
2 ∂ n 2 2 dτ 2iS h¯ L n × + ρs ∇ n + χ⊥ S L . ∂τ
0
is simply a Gaussian and can be carried out that yields the action for a (d + 1)The functional integral over L dimensional non-linear sigma model SAFM =
β d r
dτ ρs
2 2 1 ∂ n n + ∇ , 2 c ∂τ
0
√ where c = ρs χ⊥ /h¯ is the spin-wave velocity. References [1] [2] [3] [4] [5] [6] [7] [8]
N.H. Ibragimov, Transformation Groups Applied to Mathematical Physics, Reidel, Boston, 1985. P. Olver, Applications of Lie Groups to Differential Equations, Springer, 1986. L.V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982. R. Balakrishnan, A.R. Bishop, Phys. Rev. B 40 (1989) 9194. I.G. Bostrem, A.S. Ovchinnikov, R.F. Egorov, Phys. Lett. A 279 (2001) 33. A.M. Kosevich, B.A. Ivanov, A.S. Kovalev, Phys. Rep. 194 (1990) 119. A.A. Belavin, A.M. Polyakov, JETP Lett. 22 (1975) 245. F.D.M. Haldane, Phys. Lett. A 93 (1983) 464; F.D.M. Haldane, Phys. Rev. Lett. 50 (1983) 1153. [9] I. Affleck, Nucl. Phys. B 257 (1985) 397; I. Affleck, Nucl. Phys. B 265 (1985) 409. [10] I. Affleck, F.D.M. Haldane, Phys. Rev. B 36 (1987) 5291. [11] S. Sachdev, cond-mat/9303014.
Physics Letters A 292 (2002) 325–334 www.elsevier.com/locate/pla
The variational symmetries and conservation laws in classical theory of Heisenberg (anti)ferromagnet R.F. Egorov, I.G. Bostrem, A.S. Ovchinnikov ∗ Department of Theoretical Physics, Ural State University, Lenin Ave. 51, Ekaterinburg 620083, Russia Received 4 June 2001; received in revised form 26 October 2001; accepted 3 December 2001 Communicated by A.R. Bishop
Abstract The non-linear partial differential equations describing the spin dynamics of Heisenberg ferro- and antiferromagnet are studied by Lie transformation group method. The generators of the admitted variational Lie symmetry groups are derived and conservation laws for the conserved currents are found via Noether’s theorem. 2002 Elsevier Science B.V. All rights reserved. PACS: 02.20.T; 75.10.J Keywords: Symmetry groups; Noether theorem; Heisenberg model
1. Introduction As it has been demonstrated previously in a number of works the symmetry methods are very efficient tool in application to differential equations in physics. A subject of an especial interest is a study of invariance properties (symmetries) of the equations with respect to local Lie groups point transformations of dependent and independent variables. The detailed discussion and applications of the symmetries of differential equations and variational problems may be found in Refs. [1–3]. The non-linear dynamics of n-dimensional (anti)ferromagnet is described by non-linear second-order partial differential equations (PDE). In our previous work it has been shown that the system of differential equations are the Euler–Lagrange equations of a certain action functional and the corresponding long-wave action has been constructed via Vainberg’s theorem (see also Appendix A). It is well known that a part of one parameter symmetry groups of these equations turns out to be their variational symmetries as well, i.e., a symmetries of the associated action functional. According to Noether theorem [1] such an invariance of the elementary action is a necessary and sufficient condition of an existence of conservation laws for the smooth solutions of the initial PDE. The layout of the Letter is as follows. By using the Noether identity we define the variational symmetries among the point Lie symmetries of the initial differential equations of a 2D-dimensional ferromagnet. After they are * Corresponding author.
E-mail address: [email protected] (A.S. Ovchinnikov). 0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 8 1 3 - 1
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deduced we establish explicit formulae for the conserved densities and density currents involved in the conservation laws both for a Heisenberg ferromagnet and antiferromagnet. Apparently, the analysis presented may be easily generalized for an arbitrary dimension n 3.
2. Dynamical invariants of 2D ferromagnet The non-linear partial differential equations in three independent variables (x 1 , x 2 , x 3 ) = (t, r) and two dependent variables u1 = θ (t, r) and u2 = ϕ(t, r) describing the dynamics of Heisenberg ferromagnet in continuum approximation can be written as follows [4]: 2 F1 ≡ −S sin θ ϕt + β ∆θ − cos θ sin θ ∇ϕ = 0, 2 2 F ≡ S sin θ θt + β 2 cos θ sin θ ∇θ ∇ϕ + sin θ ∆ϕ = 0, (1) where β = J S 2 /h¯ . Here and throughout the following notations are taken: α have the range 1, 2 meaning θ and ϕ, correspondingly, the usual summation convention over a repeated index is employed, θj1 ...jk (or ϕj1 ...jk ) denote the kth-order partial derivatives of the dependent variables: ∂kθ . ∂x j1 ∂x j2 . . . ∂x jk The J denotes a multi-index J = (j1 , j2 , . . . , jk ) with jk = 1, . . . , 3 (k 0) pointing to the independent variables with respect to which one differentiate. Consider a local one-parameter Lie group of point transformations acting on a space Ω of the independent and dependent variables involved in the basic equations (1). The infinitesimal generator of such a group is a vector field Xˆ on Ω, θ(J ) = θj1 ...jk =
∂ ∂ ∂ + η2 , Xˆ = ξ k k + η1 ∂x ∂θ ∂ϕ
(2)
whose components ξ k and ηα are supposed to be functions of class C∞ on Ω. The infinitesimal invariance criterion of the equations F = (F1 , F2 ) under the group G is given by the determining equations Yˆ F F=0 = 0, (3) α Yˆ = Dˆ (J ) W ∂uα(J ) , (4) where Dˆ k means total differentiation, Dˆ k = ∂k + uαk(J ) ∂uα(J ) ,
(5)
ˆ and W α = ηα − ξ i uαi are the characteristics of the canonical vector field Yˆ which is equivalent to X: Yˆ = Xˆ − ξ k Dˆ k .
(6)
The determining equations (3) are realized as an over-determined system of linear homogeneous partial differential equations with respect to the unknown functions ξ k and ηα . In the determining equations the uα(J ) up to second-order derivatives are the independent variables and these equations must be fulfilled identically over them on the manifold F. A sufficient condition of an existence of conservation laws is an invariance of the elementary action associated with the Euler equations under a symmetry group of these equations. The starting point is the so-called Noether identity [1]: Yˆ + Dˆ i · ξ i = W α Eˆ α + Dˆ i Nˆ i .
(7)
R.F. Egorov et al. / Physics Letters A 292 (2002) 325–334
In Eq. (7) Nˆ i are the Noether operators given by the expressions Nˆ i = ξ i + Dˆ (s) W α (−1)r Dˆ (r) ∂uα(s)(r) ,
327
(8)
and Eˆ α = (−1)s Dˆ (s) ∂uα(s)
(9)
are the Euler–Lagrange operators. The variational symmetries on the Euler–Lagrange equations may be found via the Noether identity Yˆ (L) + Dˆ i · ξ i L = 0 (10) with the Lagrangian density L [5]:
1 2 2 1 2 L = L(θ, ϕt , θk , ϕk ) = S(cos θ − 1)ϕt − β sin θ ∇ϕ + ∇θ . 2 2
The dot in Eq. (7) means the differentiation rule Dˆ i · ξ i (L) ≡ LDˆ i (ξ i ) + ξ i Dˆ i (L). Substituting (11) into (10) one obtains Yˆ (L) = W 1 ∂θ L + Dˆ t W 2 ∂ϕt L + Dˆ k W 1 ∂θk L + Dˆ k W 2 ∂ϕk L,
(11)
(12)
where 2 β sin 2θ ∇ϕ , ∂ϕt L = S(cos θ − 1), 2 ∂ϕk L = −β sin2 θ ϕk .
∂θ L = −S sin θ ϕt − ∂θk L = −βθk ,
After a little manipulation one gets the following equation for the unknowns ξ k and ηα : 2 β 1 + S(cos θ − 1) Dˆ t η2 − ϕt Dˆ t ξ t − ϕk Dˆ t ξ k η −S sin θ ϕt − sin 2θ ∇ϕ 2 1 ˆ − βθk Dk η − θt Dˆ k ξ t − θl Dˆ k ξ l − β sin2 θ ϕk Dˆ k η2 − ϕt Dˆ k ξ t − ϕl Dˆ k ξ l 2 β 2 t β + S(cos θ − 1)ϕt − sin2 θ ∇ϕ Dˆ t ξ + Dˆ k ξ k = 0. − ∇θ 2 2
(13)
(14)
Hereinafter, we keep the upper index k just for the space coordinates, the corresponding time component will be written explicitly. The coefficients at the derivatives of the functions θ and ϕ constitute an over-determined system of linear homogeneous partial differential equations with respect to the unknowns ξ i and ηα . For example, extracting the terms at the independent variable θt and setting equal them to zero, 2 1 2 t 1 S ηθ2 − ϕk ξθk (cos θ − 1) + βθk ξkt + θk ξθt + ϕk ξϕt + β − sin2 θ ∇ϕ (15) − ∇θ ξθ = 0, 2 2 one obtains the conditions ηθ2 = ξθk = ξkt = ξϕt = ξθt = 0
(16)
that immediately imply ξ t = ξ t (t). The results of the full analysis of Eq. (14) are summarized in Table 1. Finally, we get ξ t = ξ t (t), ξ k = ξ k (r ), η1 = η1 (t, θ ), η2 = η2 (ϕ). The unknowns ξ k satisfy to Killing’s equation ξkl + ξlk = σ δlk . As is well known, it has a solution for an arbitrary σ just for n 3. However, for the system considered there is a solution in 2D case. Indeed, from Table 1 one have σ = 2ξkk = 2ηθ1 + ξtt + div ξ ,
σ = 2ξkk = 2η1 cot θ + ξtt + div ξ + 2ηϕ2
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Table 1 Equations for the unknowns ξ k and ηα Independent variable
Equation ηt2 = 0 (cos θ − 1)(ηϕ2 + div ξ ) = η1 sin θ −S(cos θ − 1)ξtk − β sin2 θ ηk2 = 0 ⇒ ξtk = ηk2 = 0 ηk1 = 0 ξ k = η1 cot θ + 1 (ξtt + div ξ ) + ηϕ2
0 ϕt ϕk θk 2 (∇ϕ)
k
2
sin2 θ (ξkl + ξlk ) = 0 ⇒ ξkl + ξlk = 0 ξkk = ηθ1 + 12 (ξtt + div ξ )
ϕk ϕl (k = l) )2 (∇θ
ξkl + ξlk = 0
θk θl (k = l)
ηϕ1 = 0
θk ϕk (∇ϕ)2 ϕk
ξϕk = 0
Table 2 Conservation laws
1 2 3 4
(no summation over k), that imply ξ = ξ (t), t
t
ξ r = a + Bˆ r,
Conservation law 2 + β (∇θ )2 = Dˆ k (βθt θk + βϕt ϕk sin2 θ ) Dˆ t β2 sin2 θ (∇ϕ) 2 Dˆ t (S(cos θ − 1)ϕk ) = Dˆ l (Tkl ) l ) = Dˆ k (& lmn x m Tnk ) (l = z) Dˆ t ([r × S(cos θ − 1)∇ϕ] Dˆ t (S(cos θ − 1)) = Dˆ k (βϕk sin2 θ )
0 T ˆ ˆ B = −B = − b
−b , 0
η1 = 0,
η2 = υ.
− b[r × ∇θ ]z The solution found results in the following expressions for the characteristics: W 1 = −ξ t θt − a ∇θ 2 t and W = υ − ξ ϕt − a ∇ϕ − b[r × ∇ϕ]z . One can derive corresponding conservation laws through Noether’s theorem procedure, Dˆ i Nˆ i (L) = 0,
(17)
under the condition Eˆ α L = 0. The Noether operators modify the Lagrangian density into a conserved quantity C i = Nˆ i (L) with a zero total divergence. Using the found variational symmetries we first construct the operators Nˆ i from formulae (8) and then calculate from (17) the corresponding conservation laws. The results are brought together in Table 2. The physical interpretation of the obtained relations is obvious. (1) The symmetry group of translations in time t → t + a yields to the energy conservation law with the 2 + (J S 2 /2)(∇θ )2 and the energy density current T = −(J S 2 θt ∇θ + energy density W = (J S 2 /2) sin2 θ (∇ϕ) 2 2 J S ϕt ∇ϕ sin θ ). and the canonical energy-momentum tensor (2) This law connects the momentum P = h¯ S(1 − cos θ )∇ϕ 2 Tkl = hLδ ¯ kl + hβθ ¯ k θl + h¯ βϕk ϕl sin θ,
(18)
and reflects the variational symmetry under space translations xl → xl + al . = [r × P ] through the angular momenta tensor under the space (3) The conservation of angular momenta L rotation around z-axis. (4) The conservation law corresponding to the transformations ϕ → ϕ + χ connects the scalar variable sin2 θ . N = S(1 − cos θ ) with the current j = β ∇ϕ
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It is useful to apply the obtained laws to some types of solutions of Eqs. (1). We restrict consideration by finite-amplitude spin-waves, dynamical soliton and skyrmion. r − ωt [6] with the dispersion h¯ ω = J S k2 cos θ0 . (1) The finite-amplitude spin-waves θ = θ0 = const and ϕ = k The last conservation law may be interpreted as a conservation of the magnon density N = S(1 − cos θ0 ). The corresponding magnon density current is j = (J S 2 /h¯ )k sin2 θ0 . The space components of the energy-momentum tensor are 2 S(1 − cos θ0 )h¯ ω + J 2S sin2 θ0 (kx2 − ky2 ) J S 2 sin2 θ0 kx ky Tˆkl = . 2 J S 2 sin2 θ0 kx ky S(1 − cos θ0 )h¯ ω − J 2S sin2 θ0 (kx2 − ky2 ) The momentum P = S(1 − cos θ0 )h¯ k is a times of the magnon density and the elementary momentum h¯ k. 2 2 2 The energy density W = (1/2)J S k sin θ0 ≈ N hω ¯ (θ0 1) has the corresponding energy density current sin2 θ0 . We note a non-additive character of energy density at finite values θ0 . T = J S 2 kω (2) For a dynamical soliton with axial symmetry one have to use the following parametrization: θ = θ (r) and ϕ = ωt [6]. The magnon density N = S(1 − cos θ (r)) has the zero magnon density current j = 0. The energymomentum tensor is 2 2 2 J S2 sin(2ωt) dθ −S(1 − cos θ (r))h¯ ω + J 2S cos(2ωt) dθ dr 2 dr Tˆkl = dθ 2 dθ 2 . J S2 J S2 sin(2ωt) −S(1 − cos θ (r)) h ω − cos(2ωt) ¯ 2 dr 2 dr The momentum P is zero, the energy density W = (J S 2 /2)(dθ/dr)2 and the energy density current T = 0. (3) For the skyrmion the following parametrization θ = 2 tan−1 (R/r) and ϕ = tan−1 (y/x) [7] is used (for simplicity we consider an unit topological charge). Then, N = 2S
r2
R2 , + R2
jr = 0,
jϕ =
J S 2 4R 2 r . h¯ (R 2 + r 2 )2
The momentum in polar coordinates of xy-plane Pr = 0,
Pϕ =
2h¯ S R 2 , r r 2 + R2
however, Tˆkl = 0. The energy density J L2z R2 = (R 2 + r 2 )2 h¯ 2 R 2 and the energy density current T = 0. W = 4J S 2
3. Dynamical invariants of 2D two-sublattice magnet In this section Lie transformation group methods will be applied to the partial differential equations describing the dynamics of 2D generic antiferromagnet (two-sublattice magnet). The established variational Lie symmetries for ferromagnet will be employed to derive antiferromagnet conservation laws revealing its important features. The action of 2D two-sublattice magnet is [5] (see also Appendix B) L=S
2
(cos θi − 1)ϕit − β −2 sin θ1 sin θ2 cos(ϕ1 − ϕ2 ) − 2 cos θ1 cos θ2 i=1
2 − sin θ1 cos θ2 sin(ϕ1 − ϕ2 ) ∇ϕ 2 1 ∇θ 1 ∇θ + cos θ1 cos θ2 cos(ϕ1 − ϕ2 ) ∇θ
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1 ∇ϕ 1 ∇ϕ 2 + sin θ1 sin θ2 cos(ϕ1 − ϕ2 ) ∇ϕ 2 + cos θ1 sin θ2 sin(ϕ1 − ϕ2 ) ∇θ 1 ∇θ 2 . + sin θ1 sin θ2 ∇θ
(19)
The time symmetry group t → t + a under which the elementary action is invariant has the infinitesimal symmetry generator Xˆ = ∂/∂t and Noether’s operators Nˆ t = 1 −
2 i=1
∂ ∂ − ϕit , ∂θit ∂ϕit 2
θit
Nˆ k = −
i=1
2 i=1
∂ ∂ − ϕit . ∂θ ik ∂ϕik 2
θit
i=1
By introducing the functions
C t = Nˆ t (L) = −β −2 sin θ1 sin θ2 cos(ϕ1 − ϕ2 ) − 2 cos θ1 cos θ2 1 ∇θ 1 ∇θ 2 − sin θ1 cos θ2 sin(ϕ1 − ϕ2 ) ∇ϕ 2 + cos θ1 cos θ2 cos(ϕ1 − ϕ2 ) ∇θ 1 ∇ϕ 1 ∇ϕ 2 + sin θ1 sin θ2 cos(ϕ1 − ϕ2 ) ∇ϕ 2 + cos θ1 sin θ2 sin(ϕ1 − ϕ2 ) ∇θ 1 ∇θ 2 + sin θ1 sin θ2 ∇θ
(20)
and C k = Nˆ k (L) = β cos θ1 cos θ2 cos(ϕ1 − ϕ2 ) + sin θ1 sin θ2 (θ1t θ2k + θ2t θ1k ) + β sin θ1 sin θ2 cos(ϕ1 − ϕ2 )(ϕ1t ϕ2k + ϕ2t ϕ1k ) + β cos θ1 sin θ2 sin(ϕ1 − ϕ2 )(θ1t ϕ2k + θ1k ϕ2t ) − β sin θ1 cos θ2 sin(ϕ1 − ϕ2 )(θ2t ϕ1k + θ2k ϕ1t )
(21)
one gets the energy conservation law Dˆ t (C t ) + Dˆ k (C k ) = 0. By the same manner one can obtain the generators and Noether’s operators of the translation group xk → xk + ak : ∂ Xˆ = k , ∂x
Nˆ t = −
2
∂ ∂ − ϕik , ∂θ it ∂ϕit 2
θik
i=1
i=1
Nˆ l = δkl −
2
∂ ∂ − ϕj k . ∂θj l ∂ϕj l 2
θj k
i=1
i=1
The momentum conservation law is 2
Dˆ t S(cos θi − 1)ϕik = Dˆ l (Tkl ),
(22)
i=1
with the tensor Tkl = Lδkl −
2 j =1
θj k ∂θjk L −
2
ϕj k ∂ϕjk L.
(23)
j =1
The conservation of angular momenta is 2 j =1
j = Dˆ k & znl x n Tlk , Dˆ t r × S(cos θj − 1)∇ϕ z
∂ ∂ Nˆ t = −& znl x n θlα α − & znl x n ϕlα α , ∂θt ∂ϕt
∂ Xˆ = & znl x n l , ∂x
∂ ∂ Nˆ k = & znk x n − & znl x n θlα α − & znl x n ϕlα α . ∂θk ∂ϕk
(24)
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331
The action is also invariant under a simultaneous rotation in both sublattices ϕ1 = ϕ1 + χ , ϕ2 = ϕ2 + χ . The generator and Noether operators are Xˆ =
2 ∂ , ∂ϕj
Nˆ t =
j =1
2 ∂ , ∂ϕj t
Nˆ k =
j =1
2 ∂ ∂ϕj k j =1
and they imply the conservation law 2
Dˆ t (S cos θj ) = β Dˆ k sin θ1 sin θ2 cos(ϕ1 − ϕ2 )(ϕ1k + ϕ2k )
j =1
− sin(ϕ1 − ϕ2 )(θ2k sin θ1 cos θ2 − θ1k sin θ2 cos θ1 ) .
(25)
4. Conclusions Lie transformation group methods have been applied to the partial differential equations describing the dynamics of ferro- and antiferromagnets and the variational symmetries of the action functional on the Euler–Lagrange equations are established. Each such symmetry gives rise to a conservation law with a conserved current. The explicit expressions for the conserved currents are found.
Acknowledgements This work was supported by the grant NREC-005 of US CRDF (Civilian Research and Development Foundation).
Appendix A Let us take the following notations: β (1) the matrix of Frechet derivatives fˆβα = s (∂f α /∂u(s))Dˆ (s) , where the sum is taken over all multi-indices s = (s1 , s2 , . . . , sk ) with sk = 1, . . . , 3 (k 0); (2) the adjoint operator of Frechet derivatives [2] β ∗ −Dˆ fˆα =
(s)
s
·
∂f α β
.
∂u(s)
One has to found a Lagrangian density L(x, u, u , . . .) that the initial set of differential equations are the Euler– Lagrange equations of the following action: S[u] =
t1
L(x, u, u , . . .) dx ≡
dt
dV L(x, u, u , . . .).
t2
From the ratio Eˆ α L = f α +
1 0
fˆ † − fˆ
α β
λuβ (x) dλ
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is seen that that sufficient condition of the solution of the problem is the self-adjoint condition of the fˆ-operator, ∂f α β
∂u(s)
∂f β s = (−1)s+r Cs+r , Dˆ (r) α ∂u(s)(r)
β or in the matrix notations fˆβα = (fˆ † )αβ = (fˆα )∗ . The straight calculation yields to
2 cos 2θ + β∆ −Sϕt cos θ − β(∇ϕ)
fˆ = Sθt cos θ + 2β cos 2θ (∇ϕ ∇θ ) + β sin 2θ ∆ϕ + S sin θ ∂t + β sin 2θ (∇ϕ ∇) and
∇) −S sin θ ∂t − β sin 2θ (∇ϕ ∇) + β sin2 θ ∆ β sin 2θ (∇ϕ
∇θ ) + β sin 2θ ∆ϕ Sθt cos θ + 2β cos 2θ (∇ϕ ∇) + S sin θ ∂t + β sin 2θ (∇ϕ . 2 β sin 2θ (∇ϕ ∇) + β sin θ ∆
2 −Sϕt cos θ − β(∇ϕ) cos 2θ + β∆ ∗ ˆ f = ∇) −S sin θ ∂t − β sin 2θ (∇ϕ
Obviously, the claimed condition is fulfilled. The Lagrangian density can be constructed via the homotopy formula [2] L[u] =
α
1 α
u (x)
f α (x, λu, λu , . . .) dλ.
0
Using the explicit form of differential equations it results in 1 L = −S(θ ϕt − ϕθt )
1 dλ λ sin λθ + βθ ∆θ
0
dλ λ 0
1 1 θ 2 2 dλ λ sin 2λθ + βϕ∆ϕ dλ λ sin2 λθ, + β ϕ ∇ϕ ∇θ − ∇ϕ 2 0
0
or, after some manipulations, in 2 β 2 β L = S(cos θ − 1)ϕt − sin2 θ ∇ϕ − ∇θ + S Dˆ t 2 2
β sin θ β ˆ ϕ + Dk θ ∇θ + 2 C0 ϕ ∇ϕ , 1− θ 2 θ
where
1 θ (θ − sin 2θ ) + sin2 θ . 4 The last two terms present total derivatives and may be dropped. C0 =
Appendix B In this appendix we show that the action employing the Lagrangian density of Eq. (19) takes the form of the action for a (d + 1)-dimensional non-linear sigma model [8–10]. The last model yields a long-wavelength, large S description of the dynamic of d-dimensional antiferromagnets in the vicinity of a ground state with collinear long-range Neel state. A more detailed discussion of the problem may be found, for example, in [11].
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333
We start with the long-wave action functional of a d-dimensional hypercubic lattice with a nearest-neighbour exchange interaction J < 0 constructed by using non-linear partial differential equations of motion of the individual spins [5], F = −S
2
t2 d r
i=1
1 dt
t1
t2 ∂ N i ∂ Ni 1 × dλ Ni − d r dt Ω|H |Ω, ∂λ ∂t h¯ t1
0
where the density of the energy is given as follows:
Ω|H |Ω = J S 2 −d sin Θ1 sin Θ2 cos(ϕ1 − ϕ2 ) − d cos Θ1 cos Θ2 1 ∇Θ 1 ∇Θ 2 − sin Θ1 cos Θ2 sin(ϕ1 − ϕ2 ) ∇ϕ 2 + cos Θ1 cos Θ2 cos(ϕ1 − ϕ2 ) ∇Θ 2 + sin Θ1 sin Θ2 cos(ϕ1 − ϕ2 ) ∇ϕ 2 1 ∇ϕ 1 ∇ϕ + cos Θ1 sin Θ2 sin(ϕ1 − ϕ2 ) ∇Θ 1 ∇Θ 2 . + sin Θ1 sin Θ2 ∇Θ Using the vectors N i (λ, t) = {sin λΘi cos ϕi , sin λΘi sin ϕi , cos λΘi }, N i (t) = Ni (1, t), the initial functional can be presented as F = −S
2
t2 d r
i=1
1 dt
t1
t2 ∂ N i J S2 ∂ Ni α α − × ∇N1 ∇N2 . dλ Ni d r dt −d N1 N2 + ∂λ ∂t h¯ α=x,y,z t1
0
In the semiclassical limit the spins will be predominantly oriented in opposite directions on the two sublattices. It allows to introduce the parametrization d 2d 2 2 + ad L n 1 − a 2d L N1 = n 1 − a L + a L, N2 = − via a continuum field of unit length n(r , t) which describes the local orientation of the Neel ordering. In the vicinity of the Neel order the n varies slowly on the scale of a lattice spacing a. The relationships include a component r , t) of the spins which is perpendicular to the local orientation of the Neel order. The continuum fields n and L L( obey the relationships = 0, n L
n 2 = 1,
2 a −2d L
satisfy the same boundary conditions in t as N (the lattice unit a which holds for all values of t and λ, and n , L will be taken unity hereinafter). By inserting the decompositions for N 1,2 in Ω|H |Ω and expanding the result in one obtains gradients and powers of L 2 2 . n − 2d L Ω|H |Ω ≈ J S 2 d − ∇ (B.1) The first term dJ S 2 presents a total derivative and may be dropped in further consideration. The Berry phase terms FB = −S
2
t2 d r
i=1
1 dt
t1
t2 2 i ∂ N ∂ϕi ∂ N i × dλ N i d r dt (cos Θi − 1) =S ∂λ ∂t ∂t
0
i=1
t1
fields: take the following form in n and L FB ≈ −2S
t2 d r
1 dt
t1
0
∂ L ∂ n ∂ n ∂ L × × dλ n + n , ∂λ ∂t ∂λ ∂t
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R.F. Egorov et al. / Physics Letters A 292 (2002) 325–334
are retained. The terms with ( where the terms of linear order in L n[∂ n /∂λ × ∂ n/∂t]) arising from the both sublattices compensate each other. It is valid for smooth spin configurations in the vicinity of Neel state. For nontrivial topological configurations this sum may be non-zero, hence the (Θ, ϕ)-representation used for the derivation of the conservation laws is more appropriate. ∂ n /∂t and ∂ n /∂λ are all perpendicular to n and have a zero triple The FB can be simplified since the vectors L, product: t2
FB = −2S
d r
1 dt
t1
∂ ∂ ∂ n ∂ n ×L + dλ n n L × . ∂t ∂λ ∂λ ∂t
(B.2)
0
The total t derivative gives zero and the total λ derivative yields a surface contribution at λ = 1. Putting together (B.1) and (B.2) one obtains the following functional: t2 2 ∂ n 1 2 2 × n + χ⊥ S L + ρs ∇ F =− d r dt 2S h¯ n L ∂t h¯ t1
with the spin-wave stiffness ρs = |J |S 2 and the transverse susceptibility χ⊥ = 2d|J |. If to replace t → iτ , the common 1/h¯ → β and multiply the functional by −i (then F → −βSAFM ), one obtains the following continuum limit path-integral for the antiferromagnet partition function: Z=
exp(−βSAFM ), D n D L
SAFM =
β d r
2 ∂ n 2 2 dτ 2iS h¯ L n × + ρs ∇ n + χ⊥ S L . ∂τ
0
is simply a Gaussian and can be carried out that yields the action for a (d + 1)The functional integral over L dimensional non-linear sigma model SAFM =
β d r
dτ ρs
2 2 1 ∂ n n + ∇ , 2 c ∂τ
0
√ where c = ρs χ⊥ /h¯ is the spin-wave velocity. References [1] [2] [3] [4] [5] [6] [7] [8]
N.H. Ibragimov, Transformation Groups Applied to Mathematical Physics, Reidel, Boston, 1985. P. Olver, Applications of Lie Groups to Differential Equations, Springer, 1986. L.V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982. R. Balakrishnan, A.R. Bishop, Phys. Rev. B 40 (1989) 9194. I.G. Bostrem, A.S. Ovchinnikov, R.F. Egorov, Phys. Lett. A 279 (2001) 33. A.M. Kosevich, B.A. Ivanov, A.S. Kovalev, Phys. Rep. 194 (1990) 119. A.A. Belavin, A.M. Polyakov, JETP Lett. 22 (1975) 245. F.D.M. Haldane, Phys. Lett. A 93 (1983) 464; F.D.M. Haldane, Phys. Rev. Lett. 50 (1983) 1153. [9] I. Affleck, Nucl. Phys. B 257 (1985) 397; I. Affleck, Nucl. Phys. B 265 (1985) 409. [10] I. Affleck, F.D.M. Haldane, Phys. Rev. B 36 (1987) 5291. [11] S. Sachdev, cond-mat/9303014.