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2D QUANTUM ANTIFERROMAGNET 189 In this way we have two spins in each AF elementary cell which are defined in different space positions l and l ′ . This circumstance is not convenient for subsequent nonlinear changes of variables. One can introduce new variables n a,b (τ,l) which are both defined at sublattice a (or at the center of the AF elementary cell). For that we pass to the Fourier image n a,b (τ,k) of the original vectors n a (τ,l) and n b (τ,l ′ ), where the momentum vector k runs over the AF Brillouin band. We can return to the space representation and consider the coordinate ρ as a continuum variable. As a result we have the definition n a,b (τ,ρ) = √ ∑ 1/N s e ik·ρ n a,b (τ,k), (5) where 2N s is the total number of sites in the space lattice. Of course, we assume periodic boundary conditions. We can now place the variable ρ on sublattice a (or in a specific center of the AF elementary cell). One can check that the Lagrangian L kin will be the same in terms of the new variables n a,b (τ,ρ) ≡ n a,b (τ,l). In the same manner one can change the measure of integration (2) and write it out in terms of n a,b (τ,l). The Hamiltonian H retains its simple form in the momentum representation. To be valid, this procedure supposes that the action can be expanded in a power series in the vectors n a,b [9]. 2.2. Invariant Lagrangian The form of the Lagrangian L kin is not invariant under rotations although the physical problem itself is invariant. The reason for this situation is explained in detail in Appendix A.1. In Appendix A.2 an invariant form of the Lagrangian L kin is proposed (84). For the two sublattices a and b it has the form k L kin =−is n a · [m a × ṁ a ] − is n b · [m b × ṁ b ]. (6) Equation (6) for L kin is valid for any choice of the unit vectors m a,b if they satisfy the conditions m 2 a,b = 1, n a,b · m a,b = 0. For the problem of the quantum AF with two sublattices we can choose the following expression for the vectors m a,b m a,b = n b,a − xn a,b √ , x = n a · n b . (7) 1 − x 2 Substituting these expressions for m a,b into Eq. (6) we get invariant forms under rotation for L kin and also for H (see (4)) L kin = B(τ,l) = is(ṅ aτl − ṅ bτl ) · [n aτl × n bτl ] , 1 − n aτl · n bτl H = Js ∑ 2 n aτl · n bτl ′, n aτl ∈ a, n bτl ∈ b. l ′ =〈l〉 (8) Now we can introduce new variables which are more convenient: Ω(τ,l) and M(τ,l). These variables realize the stereographic mapping of a sphere n a,b = ±Ω(1 − M2 /4) − [Ω × M] , (9) 1 + M 2 /4

190 BELINICHER AND PROVIDÊNCIA where Ω 2 = 1 and Ω · M = 0. In terms of these variables the total Lagrangian L M = L kin + H has the final form L kin = 2 is ˙Ω · M 1 + M 2 /4 , H = ∑ Js2 H ll ′, l ′ =〈l〉 (10) H ll ′ ={Ω · Ω ′ [(1 − M 2 /4)(1 − M ′2 /4) − M · M ′ ] + Ω · M ′ Ω ′ · M}(1 + M 2 /4) −1 (1 + M ′2 /4) −1 , where Ω ≡ Ω τl , Ω ′ ≡ Ω τl ′, M ≡ M τl , and M ′ ≡ M ′ τl . This Lagrangian determines the quadratic Lagrangian and nonlinear terms which give the amplitudes of the magnon scatering in the tree approximation which is valid in the lowest order in 1/2s. After this change of variables the measure of integration Dµ(n) (2) becomes ∏ τl (2s + 1) 2 (1 − M 2 /4) 2π 2 (1 + M 2 /4) 3 δ(Ω 2 − 1)δ(Ω · M) dΩ dM, (11) where the index l runs over the AF (double) lattice cells. 2.3. Gauge Transformation The variable Ω is responsible for the AF fluctuations and the variable M for the ferromagnetic ones. The ferromagnetic fluctuations are small according to the smallness parameter 1/2s and therefore one can expand the Lagrangian L ΩM (10) in powers of M. The vector of the ferromagnetic fluctuations M plays the role (up to a factor 2s) of the canonical momentum conjugate to the canonical coordinate Ω. The term, in the Berry phase, of first order in M coincides (after change of variables), with previous results of Haldane [9] (see also [2–4]). From Eq. (10) one can easily extract the quadratic part, L quad , of the total Lagrangian, L quad = 2 is (M · ˙Ω) + Js 2 ∑ l ′ [Ω 2 − Ω · Ω ′ + M 2 + M · M ′ ]. (12) We stress again that this quadratic form is very close to the 1988 result of Haldane [9], but is different to the extent that Haldane’s result involves the variable L = [M × Ω]. As a result we have a quite simple quadratic form. The Lagrangian L quad (12) is very simple but the measure Dµ(n) (11) is not simple due to the presence of two delta functions. Therefore we cannot simply perform the Gaussian integration over the fields Ω and M. To solve this problem we shall use the method of Lagrange multipliers together with the saddle point approximation [14, 15] to eliminate the delta function δ(Ω 2 − 1), ∫ ∞ δ(Ω 2 − 1) = (2π) −1 exp(A λ ) dλ, −∞ −A λ = L λ = (iλ + c µ )(Ω 2 − 1), (13) where λ is a Lagrange multiplier and c µ is a constant which will be fixed with the help of the saddle point condition [14, 15]. To eliminate the delta function δ(Ω · M) we shall use a form of the Faddev–Popov trick which was proposed in [20]. Let us consider the integral

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